please what is the answer for the question below

Question

If f’(x) and g’(x) exist and f’(x) > g’(x) for all real x, then the graph of y = f(x) and the graph of y = g(x).

From the following choose the correct answer to the question above:

A: Intersect exactly once.

B: Intersect no more than once.

C: Do not intersect.

D: Could intersect more than once

Answer 1

The answer may be clearer if you define

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

and rephrase the question by asking about the number of roots of h(x) given that h'(x) > 0.

A: Not necessary, for example, exponentials have no roots.

B: Yes.

By the Mean Value Theorem, two distinct roots would imply a point where h'(a) = 0.

And A above shows that no roots are also possible.

C: Not necessarily, for example, h(x) = x does have a root.

D: No, see B above.