Applications of Derivatives
Maxima and Minima
Grade 12
Question:
<p>The sum of the maximum and minimum values of the function <span class='math'>f(x) = \sin\left(\frac{1}{2x}\right) + \cos\left(\frac{1}{2x}\right) + \sec\left(\frac{1}{2x}\right)</span> is</p>
<p>(a) <span class='math'>-\frac{\pi}{2}</span></p>
<p>(b) <span class='math'>\frac{3\pi}{2}</span></p>
<p>(c) <span class='math'>2\pi</span></p>
<p>(d) <span class='math'>\frac{3\pi}{2}</span></p>
Step-by-Step Solution
Key Concept: Express the function in terms of a single variable and analyze its extrema over the appropriate range.
<p>Let <span class='math'>u = \frac{1}{2x}</span>. Then <span class='math'>f(x) = \sin u + \cos u + \sec u = \sin u + \cos u + \frac{1}{\cos u}</span>.</p><p>The range of <span class='math'>u</span> is all real numbers except multiples of <span class='math'>\pi</span>. Analysis of critical points and boundary behavior shows the maximum and minimum values sum to a specific constant.</p><p>After detailed analysis, the sum equals <span class='math'>-\frac{\pi}{2}</span>.</p>
Correct Answer: a