The function \( f:R\to R \) is defined by \( f(x)=2x^3+5 \), then which of the following statements are correct?
(A) \( f(x) \) has no local maximum value.
(B) \( f(x) \) has no local minimum value.
(C) \( f(x) \) has both local maximum and local minimum values.
(D) \( f(x) \) has neither a local maximum value nor a local minimum value
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (D) only
\( \displaystyle f'(x)=6x^2 \)
f'(x) = 0 only at x = 0, and \( f'(x)\gt 0 \) on both sides of 0, so f'(x) does not change sign there. Hence x = 0 is not a point of local maximum or local minimum (it is a point of inflexion), and f is increasing on R.
So f has no local maximum value and no local minimum value.
Checking the statements
(A) f(x) has no local maximum value. — Correct. f' does not change sign at x = 0, its only critical point.
(B) f(x) has no local minimum value. — Correct. f' does not change sign at x = 0, its only critical point.
(C) f(x) has both local maximum and local minimum values. — Incorrect. It has neither, since \( f'(x)=6x^2\ge 0 \) everywhere.
(D) f(x) has neither a local maximum value nor a local minimum value — Correct. f is increasing on R.