Question

For all x in [1, 2]

Let f''(x) of a non-constant function f(x) exist and satisfy |f''(x)| ≤ 2. If f(1) = f(2), then

• A. There exist some a ∈ (1, 2) such that f'(a) = 0
• B. f(x) is strictly increasing in (1, 2)
• C. There exist atleast one c ∈ (1, 2) such that f'(c) > 0
• D. |f'(x)| < 2 ∀ x ∈ [1, 2]

Answer 1

Answer: (A), (C)

A, C

Answer 2

(A) Rolle's Theorem applies directly as $f(1)=f(2)$ and $f$ is differentiable.

(C) If $f'(x) \leq 0$ everywhere, $f(1)=f(2)$ would imply $f$ is constant, contradicting the given information. Thus, $f'(x)$ must be positive somewhere.

(B) $f'(a)=0$ contradicts $f$ being strictly increasing.

(D) It can be shown that $|f'(x)| \leq 2$. A specific construction shows that $|f'(x)|$ can be equal to 2, so strict inequality does not hold.