Question

The greatest value of ${\sin }^{3}x+{\cos }^{3}x$ in $[0,\frac{\pi }{2}]$ is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (A)

1

$f\left(x\right)=si{n}^{3}x+co{s}^{3}x$
$f^{\prime }\left(x\right)=3sinxcosx(sinx-cosx)$
For maxima or minma,
$f^{\prime }\left(x\right)=0$
$\Rightarrow sinx=0;cosx=0;sinx-cosx=0$
$\Rightarrow x=0,\frac{\pi }{2},\frac{\pi }{4}$
$f^{\prime \prime }\left(\frac{\pi }{4}\right)>0$
So, f(x) has a minimum at $x=\frac{\pi }{4}$
$f\left(0\right)=1$ and $f\left(\frac{\pi }{2}\right)=1$
So, the maximum value of $f\left(x\right)$ is 1.