# Math Help - Proving an inequality using the mean value theorem

1. ## Proving an inequality using the mean value theorem

The question:
By using the mean value theorem, show that $1 + x < e^x$ whenever x > 0.

My attempt:
I took $e^x$ as the function, and [0,x] as the interval.

$\frac{f(x) - f(0)}{x} = e^c$

$\frac{e^x - 1}{x} = e^c$

e^c > 1 (because c must be in the interval (0, x), and e^0 = 1)

$\frac{e^x - 1}{x} > 1$
$e^x - 1 > x$ //we can do this since we know x is positive
$e^x > x + 1$
Therefore $x + 1 < e^x$

Is this correct? Thanks!

2. Completely correct!

3. Awesome! So I'm not completely hopeless after all. :P

Thanks.