let $g$ be the function given by $g(x)=x^3e^{kx}$, where $k$ is a constant. For what value of $k$ does $g$ have a critical point at $x=\dfrac{1}{2}$ ?

Summary Submit Skip Question

Use the product rule to obtain the first derivative.

$$g(x)=x^3e^{kx}$$ $$g'(x)=x^3(ke^{kx})+3x^2(e^{kx})$$ $$g'(x)=x^2e^{kx}(kx+3)$$

$g'(x)=0$ when $kx+3=0$

$$kx+3=0$$

At the point $x=\dfrac{1}{2}$,

$$\frac{1}{2}k+3=0$$ $$k = \boxed{-6}$$