Chapter 6.3, Problem 10E

True of False: Suppose a function is continuous on [a, b] where a < b and differentiable on (a, b) can this function have these three properties f(a) = 0, f(b) = 0, and f′(x) > 0.

Find all differentiable functions f: R→R such that f'(x) = F(x + n) – f(x) n for all real numbers x and all positive integers n.

Try to prove the following by using mean value theorem

Determine whether the statement is true or false. There exists a function f such that f(x) > 0, f'(x) > 0, and f"(x) < O for all x. True False

A function is said to be Hoelder a continuous on [a, b] iff |f(x) – f(x')| 0. Show that if a > 1 then f is constant in [a, b).

Let f(x) be a function which is twice differentiable with f"(x) > 0 on the interval [a 1, a + 1] and f(a) = 0. Prove that there exists some c ER with le-a| f(a-1) + f(a + 1) 2

Consider the function f: RR defined by f(x) = 0 if x is irrational and f(x) = 1 if x is rational Is f integrable over [0, 1] (or any other bounded and closed interval)?

True or false?

Let f(x) be a function on the closed interval [-2, 2] as given in the following table. f(x) -2 5 -1 -3 -2 -1 2 Based on the intemediate value theorem, the interval that contain the solution to the fo) is [-3, -2]. True False