A potential energy function VVV is defined for all real values of xxx as
V(x)=x4+8x3 V(x) = x^4 + 8x^3 V(x)=x4+8x3The function has exactly two stationary points, at x=0x = 0x=0 and x=−6x = -6x=−6.
(i) Find V′′(x)V''(x)V′′(x).
(ii) Determine the nature of the stationary points. Fully justify your answer.
State the range of values of xxx for which V(x)=x4+8x3V(x) = x^4 + 8x^3V(x)=x4+8x3 is an increasing function.
A second potential function WWW is defined for all real values of xxx as
W(x)=x4−8x3 W(x) = x^4 - 8x^3 W(x)=x4−8x3(i) State the single transformation which maps the graph of VVV onto the graph of WWW.
(ii) State the range of values of xxx for which WWW 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.