how to prove that $\exp(x)\geq x+1$

Question

• Hey I want the easiest method to prove $\exp(x)\geq x+1$.

The only method I use is to consider a new function $F$, that realizes $F(x)=\exp(x)-x-1$ then calculate the derivative then use its monotony to prove that $F(x)<0$ I'm only a high school student, if you could use function study it would be easier for me to understand.
So do you have any better method? this one takes me some time to write down I would love a easier method

Answer 1

$$\exp(x) = 1 + \int_0^{x} \exp(t) \ \mathrm dt > 1 + \int_0^x 1 \ \mathrm dt = 1 + x$$

Answer 2

Defining $$f(x)=e^x-x-1$$ then $$f'(x)=e^x-1$$ and $$f''(x)=e^x>0$$ Can you proceed?