Question

If $f\left(x\right)=10\cos x+(13+2x)\sin x,$ then which of the following is/are true?

• A. $f^{\prime }\left(0\right)=0$
• B. $f^{\prime \prime }\left(x\right)+f\left(x\right)=4\cos x$
• C. $f^{\prime \prime }\left(0\right)=-6$
• D. $f^{\prime }\left(0\right)=10$

Answer 1

Answer: (B), (C)

$f^{\prime \prime }\left(0\right)=-6$

$f\left(x\right)=10\cos x+(13+2x)\sin x\Rightarrow f\left(x\right)=10\cos x+13\sin x+2x\sin x$
Differentiating w.r.t $x$, we get
$\Rightarrow f^{\prime }\left(x\right)=-10\sin x+13\cos x+2\sin x+2x\cos x\therefore f^{\prime }\left(x\right)=-8\sin x+13\cos x+2x\cos x$
Differentiating w.r.t $x$, we get
$\Rightarrow f^{\prime \prime }\left(x\right)=-8\cos x-13\sin x+2(\cos x-x\sin x)$
$\therefore f^{\prime \prime }\left(x\right)=-6\cos x-13\sin x-2x\sin x$
So,
$f^{\prime }\left(0\right)=13,f^{\prime \prime }\left(0\right)=-6f^{\prime \prime }\left(x\right)+f\left(x\right)=4\cos x$