Proving Upper Bound on Negative Exponential

Question

I was reading a paper when I came across this lemma.

$$\forall x \geq 0,\ \exp(-x) \leq 1 – x + \frac{1}{2}x^2$$

Visually, this seems to be true:

Can someone provide a succinct proof, preferably simple or using Taylor Series? Thanks 😊

Answer 1

We have: $f(x) = e^{-x} - 1+x -\dfrac{x^2}{2}\implies f'(x) = -e^{-x}+1-x\implies f''(x)= e^{-x} -1\le 0\implies f'(x) \le f'(0)= 0\implies f(x) \le f(0)=0$, and the result follows.