Question

Let g be the function given by g(x)=x2ekx, where k is a constant. For what value of k does g have a critical point at x=23?.

Answer 1

It looks like we have

$g(x) = x^2 e^{kx}$

If x = 23 is a critical point of g(x), then

$g'(x) = 2x e^{kx} + kx^2 e^{kx} = (kx^2 + 2x) e^{kx}$

$\implies g'(23) = (23^2k + 46) e^{23k} = 0$

$e^{23k}$ is positive for all k, so we're left with

23² k + 46 = 0

23² k = -46

k = -46/23²

k = -2/23