I don't know how to use mean value theorem to show that
e^x >= 1+x+ 1/2 x^2 for x>0
I hope someone can give me some advises.
Really thanks.
You don't really need the mean value theorem here (though you do need basic knowledge of Taylor Series), because it can be shown that
$\displaystyle \displaystyle e^x = \sum_{n = 0}^{\infty}\frac{1}{n!}x^n$
$\displaystyle \displaystyle = 1 + x + \frac{1}{2}x^2 + \frac{1}{3!}x^3 + \frac{1}{4!}x^4 + \dots$
$\displaystyle \displaystyle \geq 1 + x + \frac{1}{2}x^2$.