Question

If \( f'(x) = a \cos x + b \sin x \), \( f'(0) = 4 \), \( f(0) = 3 \), and \( f\left( \frac{\pi}{2} \right) = 5 \), then \( f(x) = \)

• A. \( 2 \cos x + 4 \sin x + 1 \)
• B. \( 4 \cos x + 2 \sin x + 1 \)
• C. \( 2 \cos x + 3 \sin x + 1 \)
• D. \( 4 \cos x + \sin x + 1 \)

Hint:

Use given initial conditions to solve constants of integration after integrating a derivative.

Answer 1

The Correct Option is A

We are given \( f'(x) = a \cos x + b \sin x \). Given: \[ f'(0) = a = 4 \Rightarrow a = 4
f'(x) = 4 \cos x + b \sin x \] Integrate to get \( f(x) \): \[ f(x) = 4 \sin x - b \cos x + C \] Given \( f(0) = 3 \Rightarrow 0 - b(1) + C = 3 \Rightarrow C = 3 + b \) Also \( f\left(\frac{\pi}{2}\right) = 5 \Rightarrow 4(1) - b(0) + C = 5 \Rightarrow C = 1 \Rightarrow b = -2 \) Thus, \( f(x) = 4 \sin x + 2 \cos x + 1 \)