Finding an Interval for Derivative Bounds

[Tomath]

Homework Statement

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).

Homework Equations

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?

 

[LCKurtz]

Homework Statement

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).

Homework Equations

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?

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.

 

[Tomath]

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