If x>0, show that 1 + x + x^2/2 < e^x < 1 + x + x^2/2 e^x
I do the easier inequality, hopefully, the second one is similar.
First we will prove that $\displaystyle 1+x < e^x$ for $\displaystyle x>0$.
Define $\displaystyle f(x) = e^x - x - 1$.
Then $\displaystyle f(0) = 0$ but $\displaystyle f'(x) = e^x - 1 > 0$ for $\displaystyle x>0$.
Therefore, $\displaystyle f$ is increasing for $\displaystyle x>0$.
But since we have $\displaystyle f(0)=0$ it must mean that $\displaystyle f(x) > 0$ for $\displaystyle x>0$.
Therefore, $\displaystyle e^x - x - 1 > 0 \implies e^x > 1+x$.
Second define $\displaystyle g(x) = e^x - \tfrac{1}{2}x^2 - x - 1$.
Then $\displaystyle g(0)=0$ but $\displaystyle g'(x) = e^x - x - 1>0$ for $\displaystyle x>0$ by above.
Therefore, $\displaystyle g$ is increasing for $\displaystyle x>0$.
But since we have $\displaystyle g(0)=0$ it must mean that $\displaystyle g(x)>0$ for $\displaystyle x>0$.
Therefore, $\displaystyle e^x > \tfrac{1}{2}x^2 + x + 1$