Applications of Derivatives
Monotonicity
Grade 12
Question:
<p><strong>Ex. 69</strong> The function \(f(x)\) for \(-a \leq x \leq a\) is</p>
<p>(a) always decreasing</p>
<p>(b) always increasing</p>
<p>(c) increasing for \((-a, 0)\) and decreasing for \((0, a)\)</p>
<p>(d) increasing for \((0, a)\) and decreasing for \((-a, 0)\)</p>
Step-by-Step Solution
Key Concept: For an odd function with even derivative, monotonicity is symmetric: behavior in positive domain mirrors the behavior in negative domain with reversed direction.
<p>Since $f'(x)$ is an even function (symmetric about the y-axis) and $f(x)$ is an odd function with $f(0) = 0$, the function increases for $(0, a)$ and decreases for $(-a, 0)$.</p>
Correct Answer: d