# Fill in the blanks in each of the following The function $f(x)=\frac{2x^2-1}{x^4},x>0$decreases in the interval _______.

Given $f(x)=\frac{2x^2-1}{x^4},x>0$

After applying the derivative, we get

$\mathrm{f}^{\prime}(\mathrm{x})=\frac{\mathrm{d}\left(\frac{2 \mathrm{x}^{2}-1}{\mathrm{x}^{4}}\right)}{\mathrm{d} \mathrm{x}}$
Apply the quotient rule and 0 is the differentiation of the constant term, so

$\\ f^{\prime}(x)=\frac{x^{4} \cdot \frac{d\left(2 x^{2}-1\right)}{d x}-\left(2 x^{2}-1\right) \cdot \frac{d\left(x^{4}\right)}{d x}}{\left(x^{4}\right)^{2}} \\ f^{\prime}(x)=\frac{1}{x^{8}}\left(x^{4} \cdot \frac{d\left(2 x^{2}-1\right)}{d x}-\left(2 x^{2}-1\right) \cdot \frac{d\left(x^{4}\right)}{d x}\right) \\ f^{\prime}(x)=\frac{1}{x^{8}}\left(x^{4} \cdot(4 x)-\left(2 x^{2}-1\right) \cdot\left(4 x^{3}\right)\right) \\ f^{\prime}(x)=\frac{1}{x^{8}}\left(4 x^{5}-\left(2 x^{2}\right) \cdot\left(4 x^{3}\right)-\left(4 x^{3}\right)\right) \\ f^{\prime}(x)=\frac{1}{x^{8}}\left(4 x^{5}-8 x^{5}-\left(4 x^{3}\right)\right) \\ f^{\prime}(x)=\frac{x^{3}}{x^{8}}\left(4 x^{2}-8 x^{2}-4\right) \\ f^{\prime}(x)=\frac{1}{x^{5}}\left(-4 x^{2}-4\right) \\ f^{\prime}(x)=\frac{-4}{x^{5}}\left(x^{2}-1\right)$
Equate this with 0 and get
$\\ f^{\prime}(x)=0 \\\Rightarrow \frac{-4}{x^{5}}\left(x^{2}-1\right)=0 \\\Rightarrow x^{2}-1=0 \\\Rightarrow x^{2}=1 \\\Rightarrow x=\pm 1$
$(-\infty,-1),(-1,0),(0,1)$and $(1, \infty)$ are the intervals formed by these two critical numbers
(i) in the interval $(-\infty,-1), f^{\prime}(x)>0$
$\therefore f(x) is increasing in (-\infty,-1)$

(ii) in the interval (-1,0), $\mathrm{f}^{\prime}(\mathrm{x}) \leq 0$
$\therefore \mathrm{f}(\mathrm{x})$ is decreasing in (-1,0)
(iii) in the interval (0,1), $\mathrm{f'}(x)>0$
$\therefore \mathrm{f}(\mathrm{x})$ is increasing in $(0,1)$
(iii) in the interval $(1, \infty), f^{\prime}(x)<0$
$\therefore \mathrm{f}(\mathrm{x})$ is decreasing in $(1, \infty)$
$f(x)=\frac{2 x^{2}-1}{x^{4}}, x>0$

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