Question

If \(f'(x)=k(\cos x-\sin x)\), \(f'(0)=5\), \(f\left(\frac{\pi }{2}\right)=19\), Then \(f(x)=\)

• A. \(14(\cos x-\sin x)+5\)
• B. \(14(\cos x+\sin x)+5\)
• C. \(5(\cos x-\sin x)+14\)
• D. \(5(\cos x+\sin x)+14\)

Hint:

Use f'(0) to find k, integrate, then use f(pi/2) to find the constant.

Answer 1

The Correct Option is D

Step 1: Compare the options
Options differ in the coefficient and the constant. The constant $k$ in $f'$ should appear as the coefficient of the trigonometric part.

Step 2: Use the conditions
Since $f'(0)=k=5$, the trigonometric part is $5(\sin x+\cos x)$, because the integral of $\cos x-\sin x$ is $\sin x+\cos x$.

Step 3: Check the constant
At $x=\dfrac{\pi}{2}$: $5(1)+c=19$ gives $c=14$, matching the fourth option.

Final Answer:
The coefficient 5 and the constant 14 match the last option. \[ \boxed{\text{(D) }5(\cos x+\sin x)+14} \]