Using derivative with apparent increasing values

[jfy4]

Homework Statement

"If [itex]f'(x_0)>0[/itex] for some point [itex]x_0[/itex] in the interior of the domain of [itex]f[/itex] show that there is a [itex]\delta>0[/itex] so that

[tex]f(x)<f(x_0)<f(y)[/tex]

whenever [itex]x_0-\delta<x<x_0<y<x_0+\delta[/itex]. Does this assert that [itex]f[/itex] is increasing in the interval [itex](x_0-\delta,x_0+\delta)[/itex]?"

Homework Equations

[tex]f'(x_0)=\lim_{x\rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}[/tex]

The Attempt at a Solution

To be honest, the only way I could get this to work is by assuming the function is increasing given that the derivative is greater than zero, but it appears that is something I am suppose to show in the first place. The direction I want to take in my mind is that I can make two chords on the function, one from [itex]f(x)[/itex] to [itex]f(x_0)[/itex] and the other from [itex]f(x_0)[/itex] to [itex]f(y)[/itex] using the given data that the derivative is greater than zero. But I'm not sure how to lump in the epsilon-delta stuff.

 

[Mark44]

Homework Statement

"If [itex]f'(x_0)>0[/itex] for some point [itex]x_0[/itex] in the interior of the domain of [itex]f[/itex] show that there is a [itex]\delta>0[/itex] so that

[tex]f(x)<f(x_0)<f(y)[/tex]

whenever [itex]x_0-\delta<x<x_0<y<x_0+\delta[/itex]. Does this assert that [itex]f[/itex] is increasing in the interval [itex](x_0-\delta,x_0+\delta)[/itex]?"

Homework Equations

[tex]f'(x_0)=\lim_{x\rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}[/tex]

The Attempt at a Solution

To be honest, the only way I could get this to work is by assuming the function is increasing given that the derivative is greater than zero, but it appears that is something I am suppose to show in the first place.

Right, you need to show this, so you can't just assume it.

The direction I want to take in my mind is that I can make two chords on the function, one from [itex]f(x)[/itex] to [itex]f(x_0)[/itex] and the other from [itex]f(x_0)[/itex] to [itex]f(y)[/itex] using the given data that the derivative is greater than zero. But I'm not sure how to lump in the epsilon-delta stuff.

Start with this,
[tex]f'(x_0)=\lim_{x\rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}[/tex]

but write it in terms of the epsilon-delta definition of a limit.

 

[jfy4]

Here is my attempt at the solution after some more time... :

First let

[tex]g(x)=\frac{f(x)-f(x_0)}{x-x_0}[/tex]

then we know that

[tex]\lim_{x\rightarrow x_0}g(x)>0.[/tex]

Then pick an [itex]\epsilon=g(x_0)/2>0[/itex], then there exists a [itex]\delta>0[/itex] such that for all [itex]x\in(x_0-\delta,x_0+\delta)[/itex], [itex]g(x)\in(g(x_0)-\epsilon,g(x_0)+\epsilon)[/itex]. Then pick an arbitrary element in the interval and note that if [itex]x<x_0[/itex], then [itex]f(x)<f(x_0)[/itex], and if [itex]y>x_0[/itex], then [itex]f(y)>f(x_0)[/itex], as demanded by the strict positive inequality. Then there exists a [itex]\delta[/itex] such that if [itex]x_0-\delta<x<x_0<y<x_0+\delta[/itex], then [itex]f(x)<f(x_0)<f(y)[/itex]. [itex]\blacksquare[/itex]