A model for the net capital, CCC, of a tech startup (in millions of pounds) after ttt years is given by
C(t)=3t4−16t3 C(t) = 3t^4 - 16t^3 C(t)=3t4−16t3The function has exactly two stationary points, at t=0t = 0t=0 and t=4t = 4t=4.
(i) Find C′′(t)C''(t)C′′(t).
(a) (ii) Determine the nature of the stationary points. Fully justify your answer.
State the range of values of ttt for which the capital C(t)C(t)C(t) is an increasing function.
A revised model, KKK, is proposed for the same startup such that
K(t)=3t4+16t3 K(t) = 3t^4 + 16t^3 K(t)=3t4+16t3(c) (i) State the single transformation which maps the graph of CCC onto the graph of KKK.
(c) (ii) State the range of values of ttt for which KKK is an increasing function.
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.