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] [itex]\rightarrow[/itex] [itex]\mathbb{R}[/itex] with f'(a) < 0 en f'(b) > 0. Let c [itex]\in[/itex] [itex]\mathbb{R}[/itex] such that f'(a) < c. Show that there exists a [itex]\delta[/itex] >0 such that for every x [itex]\in[/itex] ]a, a + [itex]\delta[/itex][ 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 [itex]\rightarrow[/itex] 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 + [itex]\delta[/itex][ from?

 

[LCKurtz]

Homework Statement

Hi

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