e":"Identify the minimum","description":"Use the second derivative test. A positive second derivative means the gradient is increasing at the stationary point, so that point is a local minimum."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y=2x^3-12x^2+18x+5$"},{"latex":"$$y=2(3)^3-12(3)^2+18(3)+5$"},{"latex":"$$y=5$"},{"latex":"$$\\boxed{P(3,5)}$"}]},"order":3,"title":"Find the ordinate","description":"Substitute the minimum's x-coordinate into the original curve. The correct ordinate earns A1. Stating the complete ordered pair earns the final A1 mark."}]}],"generatedAt":"2026-07-20T16:32:08.369Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"03d386e1-ca09-47e7-b24a-eca1fba8c4bb","label":"b","content":"Justify that $P$ is a minimum point.","markscheme":"- $\\frac{d^2y}{dx^2} = 12x - 24$ M1\n- At $x=3, \\frac{d^2y}{dx^2} = 12 > 0 \\therefore$ minimum A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Finds second derivative $12x - 24$.","code":"M"},{"id":"p2","marks":1,"description":"Substitutes $x=3$ and concludes minimum as value is positive.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-12T02:14:06.631Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"second_derivative_test","label":"Use the second derivative","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\frac{dy}{dx}=\\htmlData{anchor=first}{\\color{@sky}{6x^2-24x+18}}$$"},{"latex":"$$$\\frac{d^2y}{dx^2}=\\htmlData{anchor=second}{\\color{@amber}{12x-24}}$$"}],"highlights":[{"color":"amber","style":"text-color-with-background","anchor":"second"}],"connections":[{"to":"second","from":"first","color":"orange","style":"solid"}]},"order":0,"title":"Find the second derivative","description":"Differentiate the curve once, then differentiate again. Obtaining the second derivative 12x - 24 earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\left.\\frac{d^2y}{dx^2}\\right|_{x=3}=\\htmlData{anchor=sub}{\\color{@sky}{12(3)-24}}$$"},{"latex":"$$$\\left.\\frac{d^2y}{dx^2}\\right|_{x=3}=\\htmlData{anchor=value}{\\color{@amber}{12>0\\;\\therefore\\;P\\text{ is a minimum point}}}$$"}],"highlights":[{"color":"amber","style":"text-color-with-background","anchor":"value"}]},"order":1,"title":"Test the point","description":"From part (a), P has x-coordinate 3. Substitute this into the second derivative. Since the second derivative at the stationary point P is positive, P is a minimum point. The substitution, positive value and conclusion earn A1."}]}],"generatedAt":"2026-07-20T16:43:42.431Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":62}},{"id":"037eaafa-64de-4214-aef0-0f0cfb011866","revisionId":"096cde78-5045-40ef-91e2-04a73c23bb95","questionNumber":"10","questionType":"LA","specification":"A curve has the equation $y = 3x^2 - 5x + 7$","difficulty":2,"difficultyBand":"medium","totalMarks":5,"sourceTitle":"Question Set 12: Differentiation","calculatorUse":null,"parts":[{"id":"f33bb9f4-3dac-4c4c-b328-4ab10f4fab48","label":"0","content":"Find the equation of the tangent to the curve at the point $P(2, 9)$. Write your answer in the form $y = mx + c$, where $m$ and $c$ are integers to be found.","markscheme":"- $\\frac{dy}{dx} = 6x - 5$ M1\n- At $x=2, m = 6(2) - 5 = 7$ A1\n- $y - 9 = 7(x - 2)$ M1\n- $y = 7x - 5$ A2","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Differentiates to find $\\frac{dy}{dx}$.","code":"M"},{"id":"p2","marks":1,"description":"Finds gradient $m = 7$ at $x = 2$.","code":"A"},{"id":"p3","marks":1,"description":"Uses $y - y_1 = m(x - x_1)$ with $(2, 9)$.","code":"M"},{"id":"p4","marks":1,"description":"Correctly expands to $y - 9 = 7x - 14$.","code":"A"},{"id":"p5","marks":1,"description":"Final answer $y = 7x - 5$.","code":"A"}],"finalAnswers":["y = 7x - 5"],"generatedAt":"2026-05-12

Differentiation

Question 9

Differentiation Questions

Differentiation

Question 9

Differentiation Questions