Sketch the graph of any cubic function that has both three distinct real roots and a negative coefficient of x3x^3x3.
The function g(x)g(x)g(x) is defined by
g(x)=x3−4ax2+k g(x) = x^3 - 4ax^2 + k g(x)=x3−4ax2+kwhere aaa and kkk are constants and a>0a > 0a>0.
Show that there is a stationary point where the curve crosses the yyy-axis.
Given that the equation g(x)=0g(x) = 0g(x)=0 has three distinct real roots, find the range of possible values for kkk in terms of aaa by considering the positions of the local maximum and local minimum points.
425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.