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Find $g(x)$ and classify the stationary point.","markscheme":"- $g(2)=10a+4b=-10$ M1\n- $g'(x)=3ax^2+2bx+a$ and $g'(2)=13a+4b=0$ M1\n- Solving gives $a=10/3$, $b=-65/6$ A1\n- $g(x)=\\frac{10}{3}x^3-\\frac{65}{6}x^2+\\frac{10}{3}x$ A1\n- $g''(2)=20(2)-65/3=55/3>0$, so it is a minimum A1","marks":5,"order":0,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"3ab032b6-02d6-41a7-9011-65b0f4e2c5b0","revisionId":"d5d79662-82df-4b4b-bf2a-d9cd13ecbd94","questionNumber":null,"questionType":"LA","specification":"A research probe measures the signal intensity $I$ at a distance $x$ from a source. The intensity is modeled by the function\n\n\\[I = 54x^4 - 485x + 500 + \\frac{32}{x^2} \\quad x \\neq 0\\]","difficulty":5,"difficultyBand":"hard","tier":null,"totalMarks":9,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"f582d1d6-c442-4165-a0f4-46f747c5a4ce","label":"a","content":"Find $\\frac{dI}{dx}$.","markscheme":"- Differentiates $54x^4$ to $216x^3$ B1\n- Differentiates $-485x$ to $-485$ B1\n- Differentiates $32x^{-2}$ to $-64x^{-3}$ B1","marks":3,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"power_rule","label":"Term-by-Term Power Rule","steps":[{"type":"text","order":0,"title":"Rewrite the fractional term","content":"Before differentiating, let's rewrite the term containing $x$ in the denominator using a negative exponent. This allows us to use the standard power rule on every term.\n\nRecall that:\n$$\\frac{1}{x^n} = x^{-n}$$\n\nApplying this to our fraction:\n$$\\frac{32}{x^2} = 32x^{-2}$$\n\nSo we can write the entire function as:\n$$I = 54x^4 - 485x + 500 + 32x^{-2}$$"},{"type":"text","order":1,"title":"Apply the power rule","content":"Now we differentiate term-by-term.\n\nThe standard power rule states that:\n$$\\frac{d}{dx}\\left(a x^n\\right) = a n x^{n-1}$$\n\nLet's apply this rule to each of our terms:\n\n- For $54x^4$:\n$$\\frac{d}{dx}(54x^4) = 54 \\times 4x^3 = 216x^3$$\nThis gives us our first term: $216x^3$ (1 mark).\n\n- For $-485x$:\n$$\\frac{d}{dx}(-485x) = -485$$\nThis gives us our second term: $-485$ (1 mark).\n\n- For the constant term $500$, the derivative is $0$.\n\n- For $32x^{-2}$:\n$$\\frac{d}{dx}(32x^{-2}) = 32 \\times (-2)x^{-3} = -64x^{-3}$$\nThis gives us our final term: $-64x^{-3}$ (1 mark)."},{"type":"text","order":2,"title":"State the final derivative","content":"Combining all of our differentiated terms gives the final derivative:\n\n$$\\frac{dI}{dx} = 216x^3 - 485 - 64x^{-3}$$\n\nAlternatively, you can write the final term back in fractional form:\n\n$$\\frac{dI}{dx} = 216x^3 - 485 - \\frac{64}{x^3}$$\n\nEither form is fully correct and will receive full marks."}]},{"id":"quotient_rule","label":"Combining Power and Quotient Rules","steps":[{"type":"text","order":0,"title":"Identify the terms","content":"To find $\\frac{dI}{dx}$, we can split the function into two parts:\n- The polynomial terms: $54x^4 - 485x + 500$\n- The fractional term: $\\frac{32}{x^2}$\n\nWe will differentiate the polynomial terms using the standard power rule and the fractional term using the quotient rule."},{"type":"text","order":1,"title":"Differentiate polynomial terms","content":"Recall the power rule:\n$$\\frac{d}{dx}(ax^n) = anx^{n-1}$$\n\nApplying this to each polynomial term:\n\n- For $54x^4$:\n$$\\frac{d}{dx}(54x^4) = 54 \\times 4x^3 = 216x^3$$\nThis earns 1 mark.\n\n- For $-485x$:\n$$\\frac{d}{dx}(-485x) = -485$$\nThis earns 1 mark.\n\n- For the constant $500$:\n$$\\frac{d}{dx}(500) = 0$$"},{"type":"hyper_latex","block":{"nodes":[{"text":"Let's find the derivative of the fractional term using the quotient rule formula:"},{"latex":"$$y = \\frac{u}{v} = \\frac{32}{x^2}$"},{"latex":"$$$\\frac{d}{dx}\\left(\\frac{u}{v}\\right) = \\frac{\\htmlData{anchor=u-prime}{u'}v - u\\htmlData{anchor=v-prime}{v'}}{v^2}$$"},{"text":"Here, we define:"},{"latex":"$$$u = 32 \\implies \\htmlData{anchor=u-deriv}{u'} = 0$$"},{"latex":"$$$v = x^2 \\implies \\htmlData{anchor=v-deriv}{v'} = 2x$$"}],"highlights":[{"color":"sky","style":"underline","anchor":"u-deriv"},{"color":"amber","style":"underline","anchor":"v-deriv"}],"connections":[{"to":"u-prime","from":"u-deriv","color":"sky","label":"substitute u'","style":"solid"},{"to":"v-prime","from":"v-deriv","color":"amber","label":"substitute v'","style":"solid"}]},"order":2,"title":"Apply the quotient rule","description":"We identify $u = 32$ and $v = x^2$ to set up the quotient rule calculation."},{"type":"text","order":3,"title":"Calculate fractional derivative","content":"Substitute $u$, $v$, $u'$, and $v'$ into the quotient rule formula:\n\n$$\\frac{d}{dx}\\left(\\frac{32}{x^2}\\right) = \\frac{(0)(x^2) - (32)(2x)}{(x^2)^2}$$\n\nSimplify the numerator and the denominator:\n\n$$\\frac{d}{dx}\\left(\\frac{32}{x^2}\\right) = \\frac{-64x}{x^4} = -\\frac{64}{x^3}$$\n\nThis earns 1 mark. Using negative index notation, this is written as $-64x^{-3}$."},{"type":"text","order":4,"title":"Combine the derivatives","content":"Now, we combine all our results to obtain the final expression for $\\frac{dI}{dx}$:\n\n$$\\frac{dI}{dx} = 216x^3 - 485 - \\frac{64}{x^3}$$\n\nOr in index notation:\n\n$$\\frac{dI}{dx} = 216x^3 - 485 - 64x^{-3}$$\n\nEither notation is completely correct."}]}],"generatedAt":"2026-06-09T05:21:19.961Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"5173510c-de57-4cf3-b7e1-c9af30d42ee4","label":"b","content":"Hence find the coordinates of the stationary points of the curve.","markscheme":"- Sets $\\frac{dI}{dx} = 0$ to get $216x^3 - 485 - \\frac{64}{x^3} = 0$ M1\n- Multiplies through by $x^3$ to obtain the quadratic in $x^3$: $216x^6 - 485x^3 - 64 = 0$ M1\n- Solves the quadratic to find $x^3 = \\frac{64}{27}$ or $x^3 = -\\frac{1}{8}$ (e.g., via factorisation $(27x^3 - 64)(8x^3 + 1) = 0$ or quadratic formula) M1\n- Finds the $x$-coordinates $x = \\frac{4}{3}$ and $x = -\\frac{1}{2}$ A1\n- Substitutes $x = \\frac{4}{3}$ into $I$ to get $I = 42$ M1\n- Substitutes $x = -\\frac{1}{2}$ into $I$ to get $I = 873.875$ (or $\\frac{6991}{8}$) A1","marks":6,"order":1,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"algebraic","label":"Algebraic Differentiation","steps":[{"type":"text","order":0,"title":"Differentiate the function","content":"To find the stationary points of the curve, we first need to find the first derivative of $I$ with respect to $x$, denoted as $\\frac{dI}{dx}$.\n\nGiven the intensity function:\n$$I = 54x^4 - 485x + 500 + 32x^{-2}$$\n\nApplying the power rule term-by-term:\n$$\\frac{dI}{dx} = 4(54)x^3 - 485 + 0 + (-2)(32)x^{-3}$$\n\n$$\\frac{dI}{dx} = 216x^3 - 485 - \\frac{64}{x^3}$$\n\nThis gives our derivative."},{"type":"text","order":1,"title":"Set the derivative to zero","content":"At stationary points, the gradient of the curve is zero, so we set $\\frac{dI}{dx} = 0$:\n\n$$216x^3 - 485 - \\frac{64}{x^3} = 0$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$216x^3 - 485 - \\frac{64}{\\htmlData{anchor=denom}{x^3}} = 0$"},{"text":"To eliminate the fraction, we multiply the entire equation by $x^3$:"},{"latex":"$$216\\htmlData{anchor=term-sixth}{x^6} - 485\\htmlData{anchor=term-cube}{x^3} - 64 = 0$"}],"highlights":[{"color":"sky","style":"box","anchor":"denom"},{"color":"sky","style":"underline","anchor":"term-sixth"},{"color":"sky","style":"underline","anchor":"term-cube"}],"connections":[{"to":"term-sixth","from":"denom","color":"sky","label":"multiply by $x^3$","style":"solid"},{"to":"term-cube","from":"denom","color":"sky","label":null,"style":"solid"}]},"order":2,"title":"Form a quadratic in $x^3$","description":"To eliminate the fraction, we multiply the entire equation by $x^3$. This gives us a quadratic equation in terms of $x^3$."},{"type":"text","order":3,"title":"Solve the quadratic equation","content":"Let $u = x^3$. The equation becomes a standard quadratic equation:\n$$216u^2 - 485u - 64 = 0$$\n\nWe can solve this by factorising:\n$$(27u - 64)(8u + 1) = 0$$\n\nSolving for $u$:\n$$27u - 64 = 0 \\implies u = \\frac{64}{27}$$\n\n$$8u + 1 = 0 \\implies u = -\\frac{1}{8}$$\n\nSince $u = x^3$, we have:\n$$x^3 = \\frac{64}{27} \\quad \\text{or} \\quad x^3 = -\\frac{1}{8}$$\n\nTaking the cube root gives our $x$-values:\n$$x = \\frac{4}{3} \\quad \\text{or} \\quad x = -\\frac{1}{2}$$\n\nThis gives us the two $x$-coordinates for our stationary points."},{"type":"text","order":4,"title":"Find the $I$-coordinate for $x = 4/3$","content":"Substitute $x = \\frac{4}{3}$ into the original equation $I = 54x^4 - 485x + 500 + \\frac{32}{x^2}$:\n\n$$I = 54\\left(\\frac{4}{3}\\right)^4 - 485\\left(\\frac{4}{3}\\right) + 500 + \\frac{32}{\\left(\\frac{4}{3}\\right)^2}$$\n\n$$I = 54\\left(\\frac{256}{81}\\right) - \\frac{1940}{3} + 500 + 32\\left(\\frac{9}{16}\\right)$$\n\n$$I = \\frac{512}{3} - \\frac{1940}{3} + 500 + 18$$\n\n$$I = -\\frac{1428}{3} + 518$$\n\n$$I = -476 + 518 = 42$$\n\nSo, the first stationary point is $\\left(\\frac{4}{3}, 42\\right)$."},{"type":"text","order":5,"title":"Find the $I$-coordinate for $x = -1/2$","content":"Substitute $x = -\\frac{1}{2}$ into the original equation:\n\n$$I = 54\\left(-\\frac{1}{2}\\right)^4 - 485\\left(-\\frac{1}{2}\\right) + 500 + \\frac{32}{\\left(-\\frac{1}{2}\\right)^2}$$\n\n$$I = 54\\left(\\frac{1}{16}\\right) + \\frac{485}{2} + 500 + 32(4)$$\n\n$$I = \\frac{27}{8} + \\frac{1940}{8} + 500 + 128$$\n\n$$I = \\frac{1967}{8} + 628 = 245.875 + 628 = 873.875$$\n\nIn fractional form:\n$$I = \\frac{6991}{8}$$\n\nSo, the second stationary point is $\\left(-\\frac{1}{2}, 873.875\\right)$."}]},{"id":"calculator","label":"Using Graphing Calculator","steps":[{"type":"text","order":0,"title":"Define the target task","content":"To find the coordinates of the stationary points of the curve representing the signal intensity, we need to locate where the gradient of the function is zero:\n\n$$\\frac{dI}{dx} = 0$$\n\nUsing a **Graphing Calculator**, we can find these stationary points directly by plotting the intensity function $I$ and using the built-in local minimum/maximum tools."},{"type":"text","order":1,"title":"Plot the intensity function","content":"Enter the intensity equation into your graphing calculator:\n\n$$Y_1 = 54x^4 - 485x + 500 + \\frac{32}{x^2}$$\n\nSet an appropriate viewing window to capture both stationary points:\n* $X_{\\text{min}} = -2$\n* $X_{\\text{max}} = 3$\n* $Y_{\\text{min}} = 0$\n* $Y_{\\text{max}} = 1000$","calculator":[{"gdc":{"expression":"54*x^4 - 485*x + 500 + 32/(x^2)","instructions":"1. Open the graphing menu.\n2. Enter the function: \$f(x) = 54x^4 - 485x + 500 + \\frac{32}{x^2}\$.\n3. Set the viewing window bounds: \$x \\in [-2, 3]\$, \$y \\in [0, 1000]\$.\n4. Plot the graph."},"ti84":{"expression":"54*X^4 - 485*X + 500 + 32/(X^2)","instructions":"1. Press **[Y=]**.\n2. In \$Y_1\$, enter: `54*X^4 - 485*X + 500 + 32/(X^2)`.\n3. Press **[WINDOW]** and set: `Xmin = -2`, `Xmax = 3`, `Ymin = 0`, `Ymax = 1000`.\n4. Press **[GRAPH]**."},"desmos":{"expression":"y = 54*x^4 - 485*x + 500 + 32/x^2","instructions":"1. Type the equation in the input bar: `y = 54x^4 - 485x + 500 + 32/x^2`.\n2. Adjust the graph settings (wrench icon) to set the axis limits: \$x\$-axis: \$[-2, 3]\$, \$y\$-axis: \$[0, 1000]\$."},"description":"Plotting the function on various platforms."}]},{"type":"text","order":2,"title":"Locate the first minimum","content":"The graph shows a local minimum to the right of the vertical axis (for $x > 0$). Use your calculator's local minimum finder tool to determine the exact coordinates of this point.","calculator":[{"gdc":{"instructions":"1. Press the key for analytical analysis (e.g., **[F5] (G-Solv)** on Casio).\n2. Select **[MIN]** to find the minimum point on the screen.\n3. Use the arrow keys to cycle to the positive minimum if there are multiple."},"ti84":{"instructions":"1. Press **[2nd] [CALC]** and select **3: minimum**.\n2. Move the cursor to the left of the minimum point and press **[ENTER]** (Left Bound).\n3. Move the cursor to the right of the minimum point and press **[ENTER]** (Right Bound).\n4. Press **[ENTER]** at the \"Guess?\" prompt."},"desmos":{"instructions":"1. Click directly on the plotted curve near its lowest point in the region \$x > 0\$.\n2. A grey point will appear at the local minimum. Hover over or click it to reveal its exact coordinates."},"description":"Locate the local minimum point."}]},{"type":"text","order":3,"title":"Analyze the first point","content":"The calculator displays the coordinates for the first stationary point as:\n\n$$x \\approx 1.33333, \\quad y = 42$$\n\nRecognising the decimal, we can write this in exact rational form:\n* $x = \\frac{4}{3}$\n* $I = 42$\n\nThis gives us our first stationary point coordinate:\n\n$$\\left(\\frac{4}{3}, 42\\right)$$"},{"type":"text","order":4,"title":"Locate the second minimum","content":"Now look to the left of the vertical axis (for $x < 0$). There is another local minimum. Use the local minimum finder tool again to determine the coordinates of this second point.","calculator":[{"gdc":{"instructions":"1. Press **[F5] (G-Solv)** and select **[MIN]**.\n2. Use the arrow keys to select the minimum point that lies in the negative \$x\$-region."},"ti84":{"instructions":"1. Press **[2nd] [CALC]** and select **3: minimum**.\n2. Position your cursor to the left of the minimum in the region \$x < 0\$ and press **[ENTER]**.\n3. Position your cursor to the right of this minimum and press **[ENTER]**.\n4. Press **[ENTER]** at the \"Guess?\" prompt."},"desmos":{"instructions":"1. Click directly on the curve in the region \$x < 0\$.\n2. Hover over or click the grey dot at the bottom of this valley to view the coordinates."},"description":"Locate the second local minimum point."}]},{"type":"text","order":5,"title":"Analyze the second point","content":"The calculator displays the coordinates for the second stationary point as:\n\n$$x = -0.5, \\quad y = 873.875$$\n\nConverting these decimals to fractions:\n* $x = -\\frac{1}{2}$\n* $I = 873.875 = \\frac{6991}{8}$\n\nThis gives us our second stationary point coordinate:\n\n$$\\left(-\\frac{1}{2}, \\frac{6991}{8}\\right) \\quad \\text{or} \\quad \\left(-0.5, 873.875\\right)$$\n\nBoth points found analytically correspond exactly to the algebraic solution points obtained by setting $\\frac{dI}{dx} = 0$."}]}],"generatedAt":"2026-06-09T05:24:08.197Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"f582d1d6-c442-4165-a0f4-46f747c5a4ce:power_rule","label":"(a) Term-by-Term Power Rule","steps":[{"type":"text","order":0,"title":"Rewrite the fractional term","content":"Before differentiating, let's rewrite the term containing $x$ in the denominator using a negative exponent. This allows us to use the standard power rule on every term.\n\nRecall that:\n$$\\frac{1}{x^n} = x^{-n}$$\n\nApplying this to our fraction:\n$$\\frac{32}{x^2} = 32x^{-2}$$\n\nSo we can write the entire function as:\n$$I = 54x^4 - 485x + 500 + 32x^{-2}$$"},{"type":"text","order":1,"title":"Apply the power rule","content":"Now we differentiate term-by-term.\n\nThe standard power rule states that:\n$$\\frac{d}{dx}\\left(a x^n\\right) = a n x^{n-1}$$\n\nLet's apply this rule to each of our terms:\n\n- For $54x^4$:\n$$\\frac{d}{dx}(54x^4) = 54 \\times 4x^3 = 216x^3$$\nThis gives us our first term: $216x^3$ (1 mark).\n\n- For $-485x$:\n$$\\frac{d}{dx}(-485x) = -485$$\nThis gives us our second term: $-485$ (1 mark).\n\n- For the constant term $500$, the derivative is $0$.\n\n- For $32x^{-2}$:\n$$\\frac{d}{dx}(32x^{-2}) = 32 \\times (-2)x^{-3} = -64x^{-3}$$\nThis gives us our final term: $-64x^{-3}$ (1 mark)."},{"type":"text","order":2,"title":"State the final derivative","content":"Combining all of our differentiated terms gives the final derivative:\n\n$$\\frac{dI}{dx} = 216x^3 - 485 - 64x^{-3}$$\n\nAlternatively, you can write the final term back in fractional form:\n\n$$\\frac{dI}{dx} = 216x^3 - 485 - \\frac{64}{x^3}$$\n\nEither form is fully correct and will receive full marks."}]},{"id":"f582d1d6-c442-4165-a0f4-46f747c5a4ce:quotient_rule","label":"(a) Combining Power and Quotient Rules","steps":[{"type":"text","order":0,"title":"Identify the terms","content":"To find $\\frac{dI}{dx}$, we can split the function into two parts:\n- The polynomial terms: $54x^4 - 485x + 500$\n- The fractional term: $\\frac{32}{x^2}$\n\nWe will differentiate the polynomial terms using the standard power rule and the fractional term using the quotient rule."},{"type":"text","order":1,"title":"Differentiate polynomial terms","content":"Recall the power rule:\n$$\\frac{d}{dx}(ax^n) = anx^{n-1}$$\n\nApplying this to each polynomial term:\n\n- For $54x^4$:\n$$\\frac{d}{dx}(54x^4) = 54 \\times 4x^3 = 216x^3$$\nThis earns 1 mark.\n\n- For $-485x$:\n$$\\frac{d}{dx}(-485x) = -485$$\nThis earns 1 mark.\n\n- For the constant $500$:\n$$\\frac{d}{dx}(500) = 0$$"},{"type":"hyper_latex","block":{"nodes":[{"text":"Let's find the derivative of the fractional term using the quotient rule formula:"},{"latex":"$$y = \\frac{u}{v} = \\frac{32}{x^2}$"},{"latex":"$$$\\frac{d}{dx}\\left(\\frac{u}{v}\\right) = \\frac{\\htmlData{anchor=u-prime}{u'}v - u\\htmlData{anchor=v-prime}{v'}}{v^2}$$"},{"text":"Here, we define:"},{"latex":"$$$u = 32 \\implies \\htmlData{anchor=u-deriv}{u'} = 0$$"},{"latex":"$$$v = x^2 \\implies \\htmlData{anchor=v-deriv}{v'} = 2x$$"}],"highlights":[{"color":"sky","style":"underline","anchor":"u-deriv"},{"color":"amber","style":"underline","anchor":"v-deriv"}],"connections":[{"to":"u-prime","from":"u-deriv","color":"sky","label":"substitute u'","style":"solid"},{"to":"v-prime","from":"v-deriv","color":"amber","label":"substitute v'","style":"solid"}]},"order":2,"title":"Apply the quotient rule","description":"We identify $u = 32$ and $v = x^2$ to set up the quotient rule calculation."},{"type":"text","order":3,"title":"Calculate fractional derivative","content":"Substitute $u$, $v$, $u'$, and $v'$ into the quotient rule formula:\n\n$$\\frac{d}{dx}\\left(\\frac{32}{x^2}\\right) = \\frac{(0)(x^2) - (32)(2x)}{(x^2)^2}$$\n\nSimplify the numerator and the denominator:\n\n$$\\frac{d}{dx}\\left(\\frac{32}{x^2}\\right) = \\frac{-64x}{x^4} = -\\frac{64}{x^3}$$\n\nThis earns 1 mark. Using negative index notation, this is written as $-64x^{-3}$."},{"type":"text","order":4,"title":"Combine the derivatives","content":"Now, we combine all our results to obtain the final expression for $\\frac{dI}{dx}$:\n\n$$\\frac{dI}{dx} = 216x^3 - 485 - \\frac{64}{x^3}$$\n\nOr in index notation:\n\n$$\\frac{dI}{dx} = 216x^3 - 485 - 64x^{-3}$$\n\nEither notation is completely correct."}]},{"id":"5173510c-de57-4cf3-b7e1-c9af30d42ee4:algebraic","label":"(b) Algebraic Differentiation","steps":[{"type":"text","order":0,"title":"Differentiate the function","content":"To find the stationary points of the curve, we first need to find the first derivative of $I$ with respect to $x$, denoted as $\\frac{dI}{dx}$.\n\nGiven the intensity function:\n$$I = 54x^4 - 485x + 500 + 32x^{-2}$$\n\nApplying the power rule term-by-term:\n$$\\frac{dI}{dx} = 4(54)x^3 - 485 + 0 + (-2)(32)x^{-3}$$\n\n$$\\frac{dI}{dx} = 216x^3 - 485 - \\frac{64}{x^3}$$\n\nThis gives our derivative."},{"type":"text","order":1,"title":"Set the derivative to zero","content":"At stationary points, the gradient of the curve is zero, so we set $\\frac{dI}{dx} = 0$:\n\n$$216x^3 - 485 - \\frac{64}{x^3} = 0$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$216x^3 - 485 - \\frac{64}{\\htmlData{anchor=denom}{x^3}} = 0$"},{"text":"To eliminate the fraction, we multiply the entire equation by $x^3$:"},{"latex":"$$216\\htmlData{anchor=term-sixth}{x^6} - 485\\htmlData{anchor=term-cube}{x^3} - 64 = 0$"}],"highlights":[{"color":"sky","style":"box","anchor":"denom"},{"color":"sky","style":"underline","anchor":"term-sixth"},{"color":"sky","style":"underline","anchor":"term-cube"}],"connections":[{"to":"term-sixth","from":"denom","color":"sky","label":"multiply by $x^3$","style":"solid"},{"to":"term-cube","from":"denom","color":"sky","label":null,"style":"solid"}]},"order":2,"title":"Form a quadratic in $x^3$","description":"To eliminate the fraction, we multiply the entire equation by $x^3$. This gives us a quadratic equation in terms of $x^3$."},{"type":"text","order":3,"title":"Solve the quadratic equation","content":"Let $u = x^3$. The equation becomes a standard quadratic equation:\n$$216u^2 - 485u - 64 = 0$$\n\nWe can solve this by factorising:\n$$(27u - 64)(8u + 1) = 0$$\n\nSolving for $u$:\n$$27u - 64 = 0 \\implies u = \\frac{64}{27}$$\n\n$$8u + 1 = 0 \\implies u = -\\frac{1}{8}$$\n\nSince $u = x^3$, we have:\n$$x^3 = \\frac{64}{27} \\quad \\text{or} \\quad x^3 = -\\frac{1}{8}$$\n\nTaking the cube root gives our $x$-values:\n$$x = \\frac{4}{3} \\quad \\text{or} \\quad x = -\\frac{1}{2}$$\n\nThis gives us the two $x$-coordinates for our stationary points."},{"type":"text","order":4,"title":"Find the $I$-coordinate for $x = 4/3$","content":"Substitute $x = \\frac{4}{3}$ into the original equation $I = 54x^4 - 485x + 500 + \\frac{32}{x^2}$:\n\n$$I = 54\\left(\\frac{4}{3}\\right)^4 - 485\\left(\\frac{4}{3}\\right) + 500 + \\frac{32}{\\left(\\frac{4}{3}\\right)^2}$$\n\n$$I = 54\\left(\\frac{256}{81}\\right) - \\frac{1940}{3} + 500 + 32\\left(\\frac{9}{16}\\right)$$\n\n$$I = \\frac{512}{3} - \\frac{1940}{3} + 500 + 18$$\n\n$$I = -\\frac{1428}{3} + 518$$\n\n$$I = -476 + 518 = 42$$\n\nSo, the first stationary point is $\\left(\\frac{4}{3}, 42\\right)$."},{"type":"text","order":5,"title":"Find the $I$-coordinate for $x = -1/2$","content":"Substitute $x = -\\frac{1}{2}$ into the original equation:\n\n$$I = 54\\left(-\\frac{1}{2}\\right)^4 - 485\\left(-\\frac{1}{2}\\right) + 500 + \\frac{32}{\\left(-\\frac{1}{2}\\right)^2}$$\n\n$$I = 54\\left(\\frac{1}{16}\\right) + \\frac{485}{2} + 500 + 32(4)$$\n\n$$I = \\frac{27}{8} + \\frac{1940}{8} + 500 + 128$$\n\n$$I = \\frac{1967}{8} + 628 = 245.875 + 628 = 873.875$$\n\nIn fractional form:\n$$I = \\frac{6991}{8}$$\n\nSo, the second stationary point is $\\left(-\\frac{1}{2}, 873.875\\right)$."}]},{"id":"5173510c-de57-4cf3-b7e1-c9af30d42ee4:calculator","label":"(b) Using Graphing Calculator","steps":[{"type":"text","order":0,"title":"Define the target task","content":"To find the coordinates of the stationary points of the curve representing the signal intensity, we need to locate where the gradient of the function is zero:\n\n$$\\frac{dI}{dx} = 0$$\n\nUsing a **Graphing Calculator**, we can find these stationary points directly by plotting the intensity function $I$ and using the built-in local minimum/maximum tools."},{"type":"text","order":1,"title":"Plot the intensity function","content":"Enter the intensity equation into your graphing calculator:\n\n$$Y_1 = 54x^4 - 485x + 500 + \\frac{32}{x^2}$$\n\nSet an appropriate viewing window to capture both stationary points:\n* $X_{\\text{min}} = -2$\n* $X_{\\text{max}} = 3$\n* $Y_{\\text{min}} = 0$\n* $Y_{\\text{max}} = 1000$","calculator":[{"gdc":{"expression":"54*x^4 - 485*x + 500 + 32/(x^2)","instructions":"1. Open the graphing menu.\n2. Enter the function: \$f(x) = 54x^4 - 485x + 500 + \\frac{32}{x^2}\$.\n3. Set the viewing window bounds: \$x \\in [-2, 3]\$, \$y \\in [0, 1000]\$.\n4. Plot the graph."},"ti84":{"expression":"54*X^4 - 485*X + 500 + 32/(X^2)","instructions":"1. Press **[Y=]**.\n2. In \$Y_1\$, enter: `54*X^4 - 485*X + 500 + 32/(X^2)`.\n3. Press **[WINDOW]** and set: `Xmin = -2`, `Xmax = 3`, `Ymin = 0`, `Ymax = 1000`.\n4. Press **[GRAPH]**."},"desmos":{"expression":"y = 54*x^4 - 485*x + 500 + 32/x^2","instructions":"1. Type the equation in the input bar: `y = 54x^4 - 485x + 500 + 32/x^2`.\n2. Adjust the graph settings (wrench icon) to set the axis limits: \$x\$-axis: \$[-2, 3]\$, \$y\$-axis: \$[0, 1000]\$."},"description":"Plotting the function on various platforms."}]},{"type":"text","order":2,"title":"Locate the first minimum","content":"The graph shows a local minimum to the right of the vertical axis (for $x > 0$). Use your calculator's local minimum finder tool to determine the exact coordinates of this point.","calculator":[{"gdc":{"instructions":"1. Press the key for analytical analysis (e.g., **[F5] (G-Solv)** on Casio).\n2. Select **[MIN]** to find the minimum point on the screen.\n3. Use the arrow keys to cycle to the positive minimum if there are multiple."},"ti84":{"instructions":"1. Press **[2nd] [CALC]** and select **3: minimum**.\n2. Move the cursor to the left of the minimum point and press **[ENTER]** (Left Bound).\n3. Move the cursor to the right of the minimum point and press **[ENTER]** (Right Bound).\n4. Press **[ENTER]** at the \"Guess?\" prompt."},"desmos":{"instructions":"1. Click directly on the plotted curve near its lowest point in the region \$x > 0\$.\n2. A grey point will appear at the local minimum. Hover over or click it to reveal its exact coordinates."},"description":"Locate the local minimum point."}]},{"type":"text","order":3,"title":"Analyze the first point","content":"The calculator displays the coordinates for the first stationary point approximately as:\n\n$$x \\approx 1.33333, \\quad y \\approx 42$$\n\nThese numerical readings do not by themselves establish the exact value $x=\\frac{4}{3}$. Algebraic verification is required: solving $\\frac{dI}{dx}=0$ gives $x=\\frac{4}{3}$, and substituting this value into the original function gives $I=42$.\n\nTherefore, the exact stationary point, after algebraic verification, is:\n\n$$\\left(\\frac{4}{3},42\\right)$$"},{"type":"text","order":4,"title":"Locate the second minimum","content":"Now look to the left of the vertical axis (for $x < 0$). There is another local minimum. Use the local minimum finder tool again to determine the coordinates of this second point.","calculator":[{"gdc":{"instructions":"1. Press **[F5] (G-Solv)** and select **[MIN]**.\n2. Use the arrow keys to select the minimum point that lies in the negative \$x\$-region."},"ti84":{"instructions":"1. Press **[2nd] [CALC]** and select **3: minimum**.\n2. Position your cursor to the left of the minimum in the region \$x < 0\$ and press **[ENTER]**.\n3. Position your cursor to the right of this minimum and press **[ENTER]**.\n4. Press **[ENTER]** at the \"Guess?\" prompt."},"desmos":{"instructions":"1. Click directly on the curve in the region \$x < 0\$.\n2. Hover over or click the grey dot at the bottom of this valley to view the coordinates."},"description":"Locate the second local minimum point."}]},{"type":"text","order":5,"title":"Analyze the second point","content":"The calculator displays the coordinates for the second stationary point approximately as:\n\n$$x \\approx -0.5, \\quad y \\approx 873.875$$\n\nThese numerical readings do not by themselves establish the exact fractional values. Algebraic verification gives $x=-\\frac{1}{2}$ and, on substituting into the original function, $I=873.875=\\frac{6991}{8}$.\n\nThus, after algebraic verification, the exact stationary point is:\n\n$$\\left(-\\frac{1}{2},\\frac{6991}{8}\\right)$$\n\nThe graphing calculator locates the points numerically; the exact coordinates require the algebraic check described 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1.6 Differentiation

Question 4

1.6 Differentiation Questions

1.6 Differentiation

Question 4

1.6 Differentiation Questions