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.