Minimum possible value of maximum function

The function $5x$ is strictly increasing for $x > 0$. The function $52 - 2x^2$ is strictly decreasing for $x > 0$.

The minimum of $\max\{5x, 52 - 2x^2\}$ occurs at the point of intersection: $5x = 52 - 2x^2$ $2x^2 + 5x - 52 = 0$ $(2x + 13)(x - 4) = 0$ Since $x > 0$, we have $x = 4$.

At $x = 4$, $f(4) = 5(4) = 20$. So the minimum possible value of $f(x)$ is 20.