Using Taylor's theorem to show that : $$\forall x>0 \qquad 1+x+\frac{x^2}{2} < e^x < 1+x+\frac{x^2}{2} e^x$$
What I see is, its starting terms resembles the expansion of maclaurin series of $e^x$, is that someway relevant?
In fact i can solve it without taylor's theorem, buy observing the signs of $f'(x)$ in various subsets of real line, I will prove the left side and then right, but I don't think this is the efficient way to do it.
By Taylor's theorem, if $f(x)=e^x$ then for some $0<t<x$, $$f(x)=f(0)+f'(0)x+\frac{f''(t)x^2}{2}$$ that is $$e^x=1+x+\frac{e^tx^2}{2}.$$ The given inequalities follow from the fact that $1=e^0<e^t<e^x$.