If maximum value of \(f(x) = 2x^3 + 3x^2 - 6ax + 10\) occurs at \(x = -3\), then the value of \(a\) is _____.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 6
A cubic has no greatest value on the whole real line, so the maximum here is the local maximum. At a local maximum the first derivative is zero:
\(f'(x) = 6x^2 + 6x - 6a\)
\(f'(-3) = 54 - 18 - 6a = 0\)
\(6a = 36\)
\(a = 6\)
Check with the second derivative:
\(f''(x) = 12x + 6\)
\(f''(-3) = -36 + 6 = -30 \lt 0\)
so \(x = -3\) is indeed a point of local maximum when \(a = 6\).