Assume that $f$ is a continuously differentiable function on $[0,\infty)$ such that
- $f$ is nonnegative and compactly supported,
- $f(0) = 0$,
- $\int_0^\infty f(x) dx = 1$.
Denote $$ a = \max_{x \ge 0} e^x f(x) $$
How small can $$b = \max_{x \ge 0} |e^x f'(x)|$$ be in terms of $a$?
Since $g(x) := e^x f(x)$ is continuous, we have $a = g(y)$ for some $y > 0$. Clearly $g'(y) = 0$, hence $$ a = g(y) = g(y) - g'(y) = e^yf(y) - e^yf(y) - e^y f'(y) = -e^y f'(y) \le \max_{x \ge 0} |e^x f'(x)| = b. $$ This shows that $b \ge a$. Is it possible to obtain sharper estimate of $b$ in terms of $a$?