The function \(f(x) = x^3 - 6x^2 + 9x + 4\) is strictly decreasing in the interval
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \((1, 3)\)
A function is strictly decreasing where its derivative is negative.
\(f'(x) = 3x^2 - 12x + 9\)
\(f'(x) = 3(x^2 - 4x + 3)\)
\(f'(x) = 3(x - 1)(x - 3)\)
For \(1 \lt x \lt 3\), the factor \(x - 1\) is positive and \(x - 3\) is negative, so \(f'(x) \lt 0\).
For \(x \lt 1\) and for \(x \gt 3\), both factors have the same sign, so \(f'(x) \gt 0\) and f is increasing there.
So f is strictly decreasing in (1, 3).