Applications of Derivatives
Monotonic Functions
Grade 12
Question:
<p>Let \(f(x) = x^3 + 6x^2 + ax + 2\). If \((-3, -1)\) is the largest possible interval for which \(f(x)\) is a decreasing function, then \(a = \)</p>
<p>(a) 3</p>
<p>(b) 9</p>
<p>(c) -2</p>
<p>(d) 1</p>
Step-by-Step Solution
Key Concept: For the largest decreasing interval, the derivative equals zero exactly at the endpoints of that interval.
<p>For $f(x)$ to be decreasing on $(-3, -1)$, we need $f'(x) \leq 0$ on this interval. $f'(x) = 3x^2 + 12x + a$. The roots of $f'(x) = 0$ must be at $x = -3$ and $x = -1$. Thus $f'(x) = 3(x+3)(x+1) = 3x^2 + 12x + 9$, so $a = 9$.</p>
Correct Answer: b