Let f and g be functions that are differentiable on R. Consider the following statements:
(A) If \(f'(x) \gt 0\) for every x in (a, b), then f is strictly increasing on (a, b).
(B) If f is strictly increasing on R, then \(f'(x) \gt 0\) for every real x.
(C) If \(f'(x) = 0\) for every x in (a, b), then f is constant on (a, b).
(D) If f and g are both strictly increasing on R, then the product f(x)g(x) is strictly increasing on R.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (C) only
The basic result on increasing functions says: if f′(x) > 0 for each x in an interval, f is strictly increasing there, and if f′(x) = 0 for each x in the interval, f is constant there. The converse of the first part is not true, and a product of increasing functions need not increase.
So (A) and (C) are parts of this result, (B) fails for \(f(x) = x^{3}\), and (D) fails for \(f(x) = g(x) = x\).
Checking the statements
(A) If \(f'(x) \gt 0\) for every x in (a, b), then f is strictly increasing on (a, b). — Correct. This is the basic result on increasing functions.
(B) If f is strictly increasing on R, then \(f'(x) \gt 0\) for every real x. — Incorrect. \(x^{3}\) is strictly increasing on R, but its derivative \(3x^{2}\) is 0 at x = 0.
(C) If \(f'(x) = 0\) for every x in (a, b), then f is constant on (a, b). — Correct. This is the constant-function part of the same result.
(D) If f and g are both strictly increasing on R, then the product f(x)g(x) is strictly increasing on R. — Incorrect. With f(x) = g(x) = x the product is \(x^{2}\), and \((-2)^{2} = 4 \gt (-1)^{2} = 1\).