Question

Consider the function $f\left(x\right)=si{n}^{5}x+co{s}^{5}x-1,xϵ(0,\frac{\pi }{2})$. Which of the followind is / are correct?

• A. f is monotonic increasing in $(0,\frac{\pi }{4})$,
• B. f is monotonic decreasing in $(\frac{\pi }{4},\frac{\pi }{2})$
• C. there exist some $cϵ(0,\frac{\pi }{2})$ for which $f^{\prime }\left(c\right)=0$
  
• D. The equation f(x) = 0 has two roots in $[0,\frac{\pi }{2}]$.

Answer 1

Answer: (C), (D)

The correct options are
C there exist some $cϵ(0,\frac{\pi }{2})$ for which $f^{\prime }\left(c\right)=0$

D The equation f(x) = 0 has two roots in $[0,\frac{\pi }{2}]$.

WE have
${}^{\prime }{f}^{\prime }(x{)}^{\prime }=5si{n}^{4}xcosx-5,co{s}^{4}xsinx=5,sinxcosx(sinx-cosx\left)\right(sinxcosx)\therefore f^{\prime }(x)=0atx=\frac{\pi }{4}.Also{f}^{\prime }(0)=f^{\prime }\left(\frac{\pi }{2}\right)=0$
Hence some $c\epsilon$ for $(0,\frac{\pi }{2})$ which f'(c)=0 (
By Rolle's Theorem) $\Rightarrow$ (C) is correct.
Also in $(0,\frac{\pi }{4})$ f is decreasing and in $(\frac{\pi }{4},\frac{\pi }{2})$ f is increasing $\Rightarrow$ minimum at $x=\frac{\pi }{4}$
As $f\left(0\right)=f\left(\frac{\pi }{2}\right)=0\Rightarrow 2roots\Rightarrow$ (D) is correct