Assume that f and g are differentiable functions defined on all of the real line. If for all x and if for some , then for all x.

Question

Assume that f and g are differentiable functions defined on all of the real line. If  for all x and if  for some  , then  for all x.

Quizplus · Accepted Answer

This is a direct application of the Mean Value Theorem for differentiation. Since f and g are differentiable functions, we can apply the Mean Value Theorem to both of them.

For some value c, we have:

f(b) - f(a) = f'(c)(b-a)

g(b) - g(a) = g'(c)(b-a)

We are given that f(b) - f(a) = g(b) - g(a). Therefore, we have:

f'(c)(b-a) = g'(c)(b-a)

Since (b-a) ≠ 0 (by assumption), we can divide both sides by (b-a) to obtain:

f'(c) = g'(c)

Since this holds for all c in the interval (a,b), it must be true that f'(x) = g'(x) for all x in the real line (by continuity of f' and g'). Now, we can integrate both sides with respect to x to get:

f(x) = g(x) + C

where C is an arbitrary constant. But since we are given that there exists a value x_0 such that f(x_0) = g(x_0), we can solve for C:

C = f(x_0) - g(x_0)

Substituting back into the equation f(x) = g(x) + C, we obtain:

f(x) = g(x) + (f(x_0) - g(x_0))

This holds for all x, so we have shown that f(x) - g(x) is a constant function.

Assume That F and G Are Differentiable Functions Defined on All