Question

Prove that the following functions are decreasing is given intervals
i) $f\left(x\right)={\tan }^{-1}x-x\left(x\in R\right)$
ii) $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x,x\in \left(0,\pi /4\right)$

Answer 1

Solution-

i) $f\left(x\right)=ta{n}^{-1}x-x\left(xϵR\right)$

$f^{\prime }\left(x\right)=\frac{1}{1+x^2}-1$

$x^2⩾0\frac{1}{1+x^2}⩽1$

$f^{\prime }\left(x\right)⩽0$ (precreasing function)

ii) $f\left(x\right)=si{n}^{4}x+co{s}^{4}xxϵ(0,\frac{\pi }{4})$

$f^{\prime }\left(x\right)=4cosxsi{n}^{3}x+4sinxco{s}^{3}x.$

$sinx<cosx\to cosxsi{n}^{3}x<sinxco{s}^{3}x$

then

$f^{\prime }\left(x\right)\le 0$ (decreasing function).