Question

The function $f\left(x\right)=\frac{2x^2-1}{x^4}$, $x>0$, decreases in the interval.

Answer 1

Given: $f\left(x\right)=\frac{2x^2-1}{x^4}$, $x>0$,
$\Rightarrow f\left(x\right)=\frac{2}{x^2}-\frac{1}{x^4},x>0$
$f^{\prime }\left(x\right)=\frac{-4}{x^3}+\frac{4}{x^5}=\frac{4(1-x^2)}{x^5}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{4(1-x)(1+x)}{x^5}$
Clearly, $f^{\prime }\left(x\right)<0,\forall xϵ(1,\infty )$.
Hence, $f\left(x\right)$ decreases in this interval
$xϵ(1,\infty )$ $\text{Given},x>0$