Applications of Derivatives
Local and Absolute Extrema
Grade 12
Question:
<p>Let <span class="math">\[f(x) = \begin{cases} x^3 + x^2 - 10x & -1 \leq x < 0 \\ \sin x & 0 \leq x < \frac{\pi}{2} \\ 1 + \cos x & \frac{\pi}{2} \leq x \leq \pi \end{cases}\]</span> then <span class="math">\(f(x)\)</span> has:</p>
<p>(a) local maximum at <span class="math">\(x = \frac{\pi}{2}\)</span></p>
<p>(b) local minimum at <span class="math">\(x = \frac{\pi}{2}\)</span></p>
<p>(c) absolute maximum at <span class="math">\(x = 0\)</span></p>
<p>(d) absolute maximum at <span class="math">\(x = -1\)</span></p>
Step-by-Step Solution
Key Concept: For piecewise functions, check critical points within each piece and at the boundary points where the function definition changes.
<p>Analyze each piece of the piecewise function. At <span class="math">$x = \frac{\pi}{2}$</span>, check continuity and derivatives from left and right. At <span class="math">$x = -1$</span>, evaluate the function value. Compare all critical points to determine local and absolute extrema.</p>
Correct Answer: a, d