1. The problem statement, all variables and given/known data(adsbygoogle = window.adsbygoogle || []).push({});

Give a graphical argument that if f(a)=g(a) and f'(x)>g'(x) for all x>a, then f(x)>g(x) for all x>a. Use the Mean Value Theorem to prove it.

2. Relevant equations

3. The attempt at a solution

I have sketched a graphical argument to show that f(x)>g(x). This is what I got:

Let h(x) = f(x) - g(x).

Then h'(x) = f'(x) - g'(x) > 0 for any x > a.

Now apply the MVT on the interval [a, x].

So.. h'(c) = (h(x) - h(a))/(x -a)

And then

[tex]h'(c) =\frac{f(x) - f(a) - g(x) + g(a)}{x - a} > 0[/tex]

[tex]h'(c) =f(x) - f(a) - g(x) + g(a) > 0[/tex]

[tex]h'(c) =f(x) - f(a) > g(x) - g(a)[/tex]

[tex]h'(c) =\frac{f(x) - f(a)}{x-a} > \frac{g(x) - g(a)}{x-a}[/tex]

==> h'(c) = f(x) > g(x)

==> f(x) > f(x) for all x>a

But I feel like this is wrong?!

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# Homework Help: If f(a)=g(a) and f'(x)>g'(x) for all x, use MVT to prove f(x)>g(x)

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