Question

The value of C which satisfies Rolle's Theorem for f(x) = sin⁴x + cos⁴x in \([0, \frac{π}{2}]\). Then C is:

• A. \(\frac{π}{5}\)
• B. \(\frac{π}{3}\)
• C. \(\frac{π}{4}\)
• D. \(\frac{π}{6}\)

Answer 1

The Correct Option is C

To find the value of \(C\) that satisfies Rolle's Theorem for \(f(x) = \sin^4x + \cos^4x\) in the interval \([0, \frac{π}{2}]\), we need to verify the requirements of Rolle's Theorem: continuity, differentiability, and equal values at the endpoints of the interval.

1. **Continuity and Differentiability**: The function \(f(x)\) is composed of sine and cosine functions which are continuous and differentiable over \([0, \frac{π}{2}]\). Therefore, \(f(x)\) is continuous and differentiable in this interval.

2. **Equal Endpoint Values**: Evaluate \(f(x)\) at the endpoints:

\(f(0) = \sin^4(0) + \cos^4(0) = 0^4 + 1^4 = 1\)

\(f\left(\frac{π}{2}\right) = \sin^4\left(\frac{π}{2}\right) + \cos^4\left(\frac{π}{2}\right) = 1^4 + 0^4 = 1\)

Since \(f(0) = f\left(\frac{π}{2}\right)\), the requirement for equal values at \(a\) and \(b\) is satisfied.

By Rolle's Theorem, there exists at least one \(C \in (0, \frac{π}{2})\) such that \(f'(C) = 0\).

3. **Differentiating \(f(x)\)**: We calculate the derivative of \(f(x)\):

\(f(x) = \sin^4x + \cos^4x\)

Use the chain rule:

\(\frac{d}{dx}\sin^4x = 4\sin^3x\cos x\)

\(\frac{d}{dx}\cos^4x = -4\cos^3x\sin x\)

Thus,

\(f'(x) = 4\sin^3x\cos x - 4\cos^3x\sin x\)

\(= 4\sin x\cos x(\sin^2x - \cos^2x)\)

\(= 2\sin 2x(1 - 2\cos^2x)\) (using \(\sin^2x - \cos^2x = -\cos 2x\) and \(1 - \cos 2x = 2\sin^2x\))

Setting \(f'(x) = 0\), we solve:

\(2\sin 2x(1 - 2\cos^2x) = 0\)

\(2\sin 2x = 0\) or \(1 - 2\cos^2x = 0\)

\(\sin 2x = 0\) implies \(2x = nπ\) for integers \(n\).

\(x = \frac{nπ}{2}\).

In \((0, \frac{π}{2})\), valid \(x\) values are \(x = \frac{π}{4}\).

Verify \(1 - 2\cos^2x = 0\):

\(2\cos^2x = 1\)

\(\cos^2x = \frac{1}{2}\)

\(x = \frac{π}{4}\) also satisfies this equation.

Therefore, the value of \(C\) where \(f'(C) = 0\) is \(\frac{π}{4}\).

Thus, the correct answer is:

\(\frac{π}{4}\)