If the function \(f(x) = x^{3} + ax^{2} + bx\) has a local maximum at x = 1 and a local minimum at x = 3, then a + b is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 3
At a point of local maximum or minimum of a differentiable function, f′(x) = 0. So x = 1 and x = 3 are the roots of
\( f'(x) = 3x^{2} + 2ax + b \)
A quadratic with leading coefficient 3 and roots 1 and 3 is
\( 3(x - 1)(x - 3) = 3x^{2} - 12x + 9 \)
Comparing coefficients:
\( 2a = -12,\ a = -6 \)
\( b = 9 \)
Check: \(f''(x) = 6x - 12\) gives f″(1) = −6 < 0 (maximum) and f″(3) = 6 > 0 (minimum), as required. So
\( a + b = -6 + 9 = 3 \)