Question

Let \( f(x) = (\cos^2 x)(a + \cos x) \). If \( f'\left(\frac{\pi}{3}\right) = 0 \), then the value of \( a \) is equal to

• A. \( \frac{\sqrt{3}}{2} \)
• B. \( \frac{3}{4} \)
• C. \( -\frac{3}{4} \)
• D. \( -\frac{3}{2} \)
• E. \( -1 \)

Hint:

Plug values early after differentiation to simplify calculations.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem requires finding the derivative of a function using the product rule and the chain rule, and then solving for an unknown constant `a` by using a given condition about the derivative at a specific point.
Step 2: Key Formula or Approach:
1. Use the product rule: \((uv)' = u'v + uv'\). Let \(u = \cos^2 x\) and \(v = a + \cos x\). 2. Find the derivatives \(u'\) and \(v'\). 3. Substitute these into the product rule formula to get \(f'(x)\). 4. Set \(f'(\pi/3) = 0\) and solve for `a`.
Step 3: Detailed Explanation:
1. Find the derivatives of u and v. Let \(u(x) = \cos^2 x\). Using the chain rule, \(u'(x) = 2\cos x \cdot (-\sin x) = -2\sin x \cos x = -\sin(2x)\). Let \(v(x) = a + \cos x\). The derivative is \(v'(x) = -\sin x\). 2. Apply the product rule. \[ f'(x) = u'v + uv' \] \[ f'(x) = (-\sin(2x))(a + \cos x) + (\cos^2 x)(-\sin x) \] 3. Substitute \(x = \pi/3\) and set to 0. We are given \(f'(\pi/3) = 0\). First, find the values of the trigonometric functions at \(x = \pi/3\):
\(\cos(\pi/3) = 1/2\)
\(\sin(\pi/3) = \sqrt{3}/2\)
\(\sin(2\pi/3) = \sqrt{3}/2\)
Now substitute these into the expression for \(f'(x)\): \[ f'(\pi/3) = \left(-\sin\left(2\frac{\pi}{3}\right)\right)\left(a + \cos\frac{\pi}{3}\right) + \left(\cos^2\frac{\pi}{3}\right)\left(-\sin\frac{\pi}{3}\right) = 0 \] \[ \left(-\frac{\sqrt{3}}{2}\right)\left(a + \frac{1}{2}\right) + \left(\left(\frac{1}{2}\right)^2\right)\left(-\frac{\sqrt{3}}{2}\right) = 0 \] \[ -\frac{\sqrt{3}}{2}\left(a + \frac{1}{2}\right) + \frac{1}{4}\left(-\frac{\sqrt{3}}{2}\right) = 0 \] \[ -\frac{\sqrt{3}}{2}a - \frac{\sqrt{3}}{4} - \frac{\sqrt{3}}{8} = 0 \] We can divide the entire equation by \(-\sqrt{3}\) (since it's non-zero): \[ \frac{a}{2} + \frac{1}{4} + \frac{1}{8} = 0 \] \[ \frac{a}{2} + \frac{2+1}{8} = 0 \] \[ \frac{a}{2} + \frac{3}{8} = 0 \] \[ \frac{a}{2} = -\frac{3}{8} \] \[ a = -\frac{6}{8} = -\frac{3}{4} \] Step 4: Final Answer:
The value of a is \(-\frac{3}{4}\).