The local maximum value of the function \(f(x) = 3x^{5} - 5x^{3}\), \(x \in R\), is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 2
The critical points are where \(f'(x) = 0\):
\( f'(x) = 15x^{4} - 15x^{2} \)
\( = 15x^{2}(x - 1)(x + 1) \)
So x = −1, 0 or 1. By the first derivative test (the factor \(15x^{2}\) is never negative):
at x = −1, f′ changes from + to −: local maximum
at x = 0, f′ stays − on both sides: neither maximum nor minimum
at x = 1, f′ changes from − to +: local minimum
The local maximum value is
\( f(-1) = -3 + 5 = 2 \)
Why the other options are incorrect
Option (1) 0 — Incorrect. This is f(0), but f′ does not change sign at x = 0, so it is not a local maximum.
Option (2) \(-2\) — Incorrect. This is f(1) = 3 − 5, the local minimum value.
Option (3) 1 — Incorrect. This is the point x = 1 of local minimum, not a value of f.