Question

The number of values of \(x\), where \(f(x) = \cos x + \cos 2x\) attains its maximum is

• A. 1
• B. 0
• C. 2
• D. infinite

Hint:

For periodic functions, if the maximum occurs at isolated points in one period, the total number over all reals is infinite.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
\(f(x)\) is periodic. Find maximum value and number of points where it occurs.

Step 2: Detailed Explanation:
\(f(x) = \cos x + \cos 2x = \cos x + 2\cos^2 x - 1 = 2\cos^2 x + \cos x - 1\).
Let \(t = \cos x\), \(t \in [-1, 1]\). \(g(t) = 2t^2 + t - 1\).
Vertex at \(t = -\frac{1}{4}\), \(g(-\frac{1}{4}) = 2(\frac{1}{16}) - \frac{1}{4} - 1 = \frac{1}{8} - \frac{1}{4} - 1 = -\frac{9}{8}\). Actually that's minimum. Check endpoints: \(t=1\), \(g(1) = 2 + 1 - 1 = 2\). \(t=-1\), \(g(-1) = 2 - 1 - 1 = 0\). So maximum = 2 at \(t=1\).
\(t = \cos x = 1 \implies x = 2n\pi\), infinite values.

Step 3: Final Answer:
Option (D) infinite.