TITLE: Differential inequation problem
QUESTION [6 upvotes]: Let $f:[0,1]→\mathbb{R}$ be a function. Suppose the function $f$ is twice differentiable, $f(0)=f(1)=0$ and satisfies $f''(x)-2f'(x)+f(x)\geq e^x$, $x\in [0,1]$.
Which of the following is true for $0<x<1$?
A) $0<f(x)<\infty$
B) $-\frac{1}{2}<f(x)<\frac{1}{2}$
C) $-\frac{1}{4}<f(x)<1$
D) $-\infty<f(x)<0$

This appeared in one of my test papers and I went blank over this. I am still out of ideas because I have never come across differential inequations. Moreover, the above inequation is second-order which isn't even in my course. How am I supposed to deal with this? 
It would be greatly appreciated is someone could provide a resource or the name of book where these kind of problems are found. Many thanks!

REPLY [2 votes]: $$(f(x)e^{-x})''\ge 1\implies \text{ concave upward }$$
$$f(0)e^{-0}=f(1)e^{-1}=0\implies \text{negative}\implies (D)$$