Q77 / 160 · AP EAMCET 2010 · Mathematics

If $f(x)=\sin x+\cos x$, then $f\left(\frac{\pi}{4}\right) f^{(i v)}\left(\frac{\pi}{4}\right)$ is equal to

  • A
    1
  • Correct answer
    B
    2
  • C
    3
  • D
    4

Solution

$f(x)=\sin x+\cos x$ $f^{\prime}(x)=\cos x-\sin x$ $f^{\prime \prime}(x)=-\sin x-\cos x$ $f^{\prime \prime \prime}(x)=-\cos x+\sin x$ $f^{\prime \prime \prime \prime}(x)=\sin x+\cos x$ So, $f\left(\frac{\pi}{4}\right)=f^{\prime \prime \prime \prime}\left(\frac{\pi}{4}\right)=\sin \left(\frac{\pi}{4}\right)+\cos \left(\frac{\pi}{4}\right)$ $=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=\frac{2}{\sqrt{2}}=\sqrt{2}$ Then, $\quad f\left(\frac{\pi}{4}\right) f^{\prime \prime \prime \prime}\left(\frac{\pi}{4}\right)=\sqrt{2} \times \sqrt{2}$ $=2$

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

Relevant Questions