Question

True or False:

If f and g are differentiable and f(x) ≥ g(x) for a < x < b, then f '(x) ≥ g'(x) for a < x < b.

This is what I tried:
I let f(x) = x^2 + 5
I let g(x) = x^2

Differentiating,
f'(x) = 2x
g'(x) = 2x

Now, when you evaluate for (a,b), you will get exact same area. So, I showed that the equal part of the inequality is true.

Then, I pick new functions:
f(x) = x^3
g(x) = x^2

Differentiating,
f'(x) = 3x^2
g'(x) = 2x

f' grows faster than g', and so it will have a greater area than g'. So, I showed that the "greater than" part of the ineqaulity is also true.

Answer 1

So, is the answer TRUE?

Answer 2

f is over g on the entire interval, but it is clear to see that g' > f' over the entire interval (it's steeper)

Answer 3

You cannot prove theorems like that with two examples. You can refute them with 1 counterexample though ;)