Sketch the graph of a cubic function that has three distinct real roots and a positive coefficient of x3x^3x3.
The vertical displacement, sss, of a mechanical component is modelled by the function
s(t)=k+15at2−2t3 s(t) = k + 15at^2 - 2t^3 s(t)=k+15at2−2t3where t≥0t \ge 0t≥0 is time, and aaa and kkk are constants with a>0a > 0a>0.
Show that the curve s(t)s(t)s(t) has a stationary point at its sss-intercept.
Given that the polynomial equation s(t)=0s(t) = 0s(t)=0, when extended to all real values of ttt, has three distinct real roots, determine the range of possible values for kkk in terms of aaa. You must use the second derivative to justify the nature of the stationary points used in your calculation.
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.