# Derivatives and deltas.

1. May 28, 2012

### Tomath

1. The problem statement, all variables and given/known data
Hi

I've been giving the following problem:
We have a differentiable function f: [a,b] $\rightarrow$ $\mathbb{R}$ with f'(a) < 0 en f'(b) > 0. Let c $\in$ $\mathbb{R}$ such that f'(a) < c. Show that there exists a $\delta$ >0 such that for every x $\in$ ]a, a + $\delta$[ the following holds:

f(x) < f(a) + c(x-a).
2. Relevant equations

3. The attempt at a solution
My attempt at a solution is the following:
Using the definition of the derivative we have the following:

lim x $\rightarrow$ a f(x) - f(a)/(x - a) < c so f(x) < f(a) + c(x-a).

My question is, where do I get the interval ]a, a + $\delta$[ from?

2. May 28, 2012

### LCKurtz

Re: Derivative's and delta's.

You have $$\frac {f(x) - f(a)}{x-a}\rightarrow f'(a)<c$$ as $x\rightarrow a$. That doesn't mean it is less than $c$ for all $x$. $x$ has to be sufficiently close to $a$. You need to think about the definition of limit to get a $\delta$ that works.

3. May 28, 2012

### Tomath

Re: Derivative's and delta's.

Okay I've figured it out. Thanks for your help ^^.