Hi,

How do we prove the answer for 9367 Q54?

If f and g are real-valued differentiable functions and if f'(x)>=g'(x) for all x in the closed interval [0,1], which of the following must be true?

(C) f(1)-g(1)>=f(0)-g(0)

Can we prove it by

$$\lim_{x\to 0}\frac{f(x)-f(0)}{x-0}\geq\lim_{x\to 0}\frac{g(x)-g(0)}{x-0}$$

$$\therefore f(x)-f(0)\geq g(x)-g(0)$$, for all x in [0,1]

$$\therefore f(1)-f(0)\geq g(1)-g(0)$$

Thanks.

## 9367 54 proof.

### Re: 9367 54 proof.

Recall that by the FTC:

$$f(1) = f(0) + \int_0^1 f'(x)dx,\,g(1) = g(0)+\int_0^1 g'(x)dx$$

That should provide the necessary ingredients to finish the proof.

Your proof is invalid because it only applies to x sufficiently close to zero.

$$f(1) = f(0) + \int_0^1 f'(x)dx,\,g(1) = g(0)+\int_0^1 g'(x)dx$$

That should provide the necessary ingredients to finish the proof.

Your proof is invalid because it only applies to x sufficiently close to zero.

### Re: 9367 54 proof.

Hmm, an alternate proof from the one above is letting h(x)=f(x)-g(x). Since differentiation is linear, h'(x) = f'(x) - g'(x). We also see that h'(x)>=0 since f'(x)>=g'(x). By the Mean Value Theorem, there exists a c in [0,1] such that h'(c) = h(1) - h(0), and h(1) - h(0)>=0. Thus, we have f(1) - g(1) - f(0) + g(0)>=0, or f(1) - f(0) >= g(1) - g(0).