Let $f:\mathbb R\rightarrow \mathbb R$ be a differentiable function. Prove that $f(x)\leq f'(x)$, then $f$ is non decreasing?
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No. Take $f(x)=-a^x$ with $1<a<e$, then $f'(x)=-a^x\log a>f(x)$, while $f(x)$ is decreasing.
By increasing, you probably mean non-decreasing (since the constant $0$ function satisfies the inequality). If $f$ is positive, then there is no problem. Otherwise, it does not seem right (c.f. Xiaochuan's counter exemple).