True or false. If true, why? If false, what is a counter example, and can you explain why it contradicts? Please

Question

If f and g are differentiable and f(x)≤ g(x) on [a,b], then f'(x)≤g'(x) on [a,b]

Answer 1

The statement is false.

A simple example is to let  g(x) = 2 for all x   (so g is a constant function) and let f(x) = x on the interval [a,b]=[0,1].

(Other choices of interval are possible using these expressions for g and f.)

 

Then f(x) ≤ g(x) on the interval, but f'(x) is a constant 1, while g'(x) is a constant 0, giving us g'(x) < f'(x).

Answer 2

FALSE:

 

Let f(x) = lnx, g(x) = x²  where 1/2 ≤ x ≤ 1

 

f(x) ≤ g(x) on [1/2, 1] since f(x) ≤ 0 and g(x) ≥ 0 on the interval

 

f'(x) = 1/x, g'(x) = 2x

 

f'(1/2) = 1/(1/2) = 2  and g'(1/2) = 2(1/2) = 1  

 

So, f(1/2) ≤ g(1/2), but f'(1/2) > g'(1/2)