# Derivative question

• Jan 8th 2012, 01:23 PM
grandunification
Derivative question
Let f be a function from R to R, and f ' (x) > f (x) for all x. Suppose f(x_0) =0. Show that f (x) > 0 for all x > x_0.

Thanks!
• Jan 9th 2012, 08:48 AM
Opalg
Re: Derivative question
Quote:

Originally Posted by grandunification
Let f be a function from R to R, and f ' (x) > f (x) for all x. Suppose f(x_0) =0. Show that f (x) > 0 for all x > x_0.

Since $f'(x_0) > f(x_0) = 0$, it follows that f is increasing at $x_0.$ Therefore there is some interval to the right of $x_0$ in which f is positive. Suppose by way of getting a contradiction that f is not positive for all $x > x_0.$ Then (by continuity, intermediate value theorem and suchlike properties) there must be a smallest point $x_1>x_0$ at which $f(x) = 0.$ Thus $f(x)>0$ whenever $x_0 But since $f(x_0) = f(x_1) = 0$ it follows from Rolle's theorem that there is a point $x_2$ with $x_0 such that $f'(x_2)=0.$ Then $f'(x_2), contradicting the fact that $f'(x)>f(x)$ for all $x>x_0.$