# Math Help - mean value theorem

1. ## mean value theorem

Use the Mean Value Theorem to prove that sqrt(1+x) < 1 + x/2 for all x > 0. Generalize this result to the function (1+x)^r, where r < 1 is a positive rational number.

Thanks!

2. Originally Posted by friday616
Use the Mean Value Theorem to prove that sqrt(1+x) < 1 + x/2 for all x > 0. Generalize this result to the function (1+x)^r, where r < 1 is a positive rational number.

Thanks!
Using the MVT, $\frac{f(x)-f(0)}{x-0}=\frac{\sqrt{1+x}-1}{x}=f'(t)$ for some $t\in[0,x]$.

$f'(t)=\frac{1}{2\sqrt{1+x}}$. For $x>0$, $f'(t)<\frac{1}{2}$.

Thus,

$\frac{\sqrt{1+x}-1}{x}<\frac{1}{2}\implies\sqrt{1+x}<1+\frac{x}{2}$ for $x>0$

as desired.

--------

Using the same method, see if you can generalize that result.