The pdf $f(x)$ of an exponential distribution with parameter $\lambda$ is
$$f(x) = \lambda \exp(-\lambda x)\quad x\geq 0$$
The cdf $F(x)$ of such a distribution is
$$F(x) = \int^x_0 f(x') dx'$$

2. In what cases do $F(x)$ and $f(x)$ intersect?
You might want to experiment with specific choices of $\lambda$ before constructing more general arguments.