If $f(x)=10 \cos x+(13+2 x) \sin x, \quad$ then $f^{\prime \prime}(x)+f(x)$ is equal to

• A

  $\cos x$
• Correct answer
  B

  $4 \cos x$
• C

  $\sin x$
• D

  $4 \sin x$

Solution

Given that
On differentiating w.r.t. $x$, we get $f^{\prime}(x)=-10 \sin x+(13+2 x) \cos x+2 \sin x$ Again differentiating, we get $\begin{aligned} f^{\prime \prime}(x)=-10 \cos x-(13+2 x) \sin x+ & 2 \cos x \\ & +2 \cos x ...(ii) \end{aligned}$ On adding Eqs. (i) and (ii), we get $f^{\prime \prime}(x)+f(x)=2 \cos x+2 \cos x=4 \cos x$

Hence, the correct answer is option: 'B'.