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.
375 exam-style questions on AQA A Level Maths 1.10 G: Differentiation, covering 1.10.1 The derivative and second derivative, 1.10.2 Differentiating standard functions, 1.10.3 Applications of differentiation, 1.10.4 Product, quotient and chain rules (A-level only), 1.10.5 Implicit and parametric differentiation (A-level only), and 1.10.6 Constructing differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.