2.15: Stationary points, second derivative test and surface area

Find the $x$-coordinates of the stationary points of the curve $y = \mathrm{e}^{3x}(2x + 3)^{6}$.

A curve has equation $y = f(x)$ and has exactly two stationary points. Given that $f''(x) = 4x - 7$, $f'(0.5) = 0$ and $f'(3) = 0$, use the second derivative test to determine the nature of each of the stationary points of this curve.

In this question all lengths are in centimetres.

The diagram shows a solid cuboid with height $h$ and a rectangular base measuring $4x$ by $x$. The volume of the cuboid is $40\text{ cm}^{3}$. Given that $x$ and $h$ can vary and that the surface area of the cuboid has a minimum value, find this value.