Unique maximizer of product of monotonically increasing and decreasing functions

Question

I need to show that the function $g(x) = (x-c)(1-F(x))$ has a unique maximizer that can be obtained by solving $\frac{dg(x)}{dx}=0$. $F(x)$ is CDF of some random variable and $c$ is a positive constant. I am assuming that the support of $F(x)$, $[x^{\text{min}}, x^{\text{max}}]$, is such that $c < x^{\text{max}}$. It seems intuitive to me that since $(x-c)$ is a monotonically non-decreasing affine function and $(1-F(x))$ is a monotonically non-increasing function, the product should keep rising and then falling after hitting a certain level. But I do not know how to rigorously prove it. I wonder if you can provide any references or hints. I appreciate any insights.