Question

Let s < t and let f:[s,t]→ℝ be a differentiable function. Suppose that f'(x) > 0 for all x

Which of the following is correct?

1. Using the definition of the derivative, it follows that f(x)<f(a) for any x<a.

2. Using Rolle's Theorem, it follows that f is a continuous function.

3. Using the Mean Value Theorem, it follows that f(t)>f(s).

4. Using Rolle's Theorem, it follows that there is some x∈[s,t] such that f′(x)=0.

5. Using the Intermediate Value Theorem, it follows that f(t)>f(s).

6. Using the definition of the derivative, it follows that f(x)>f(a) for any x>a.

Answer 1

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