Question 142
Let $\mathcal{F}$ be the set of continuous functions $f : \mathbb{R} \to \mathbb{R}$ which satisfy the condition
$$
e^{f(x)} + f(x) \geq x + 1,$$
for any real number $x$. Find the minimum value attained by the integral
$$
I(f) = \int_{0}^{e} f(x) \, dx,
$$
for $f$ lying in $\mathcal{F}$.
$$
e^{f(x)} + f(x) \geq x + 1,$$
for any real number $x$. Find the minimum value attained by the integral
$$
I(f) = \int_{0}^{e} f(x) \, dx,
$$
for $f$ lying in $\mathcal{F}$.