Question

If $f'(x)=k(\cos x-\sin x)$, $f'(0)=3$, $f\!\left(\dfrac{\pi}{2}\right)=15$, then $f(x)=$

• A. $3(\sin x+\cos x)+12$
• B. $3(\sin x+\cos x)-12$
• C. $-3(\sin x+\cos x)-12$
• D. $12(\sin x+\cos x)+3$

Hint:

Always use given initial or boundary conditions to find constants of integration.

Answer 1

The Correct Option is A

Step 1: Using the given condition $f'(0)=3$.
\[ f'(0)=k(\cos0-\sin0)=k(1-0)=k \] \[ k=3 \]
Step 2: Writing $f'(x)$.
\[ f'(x)=3(\cos x-\sin x) \]
Step 3: Integrating to find $f(x)$.
\[ f(x)=3(\sin x+\cos x)+C \]
Step 4: Using the condition $f\!\left(\dfrac{\pi}{2}\right)=15$.
\[ 15=3(1+0)+C \Rightarrow C=12 \]
Step 5: Conclusion.
\[ f(x)=3(\sin x+\cos x)+12 \]