Find the interval for which f(x)=x^4-2 x^2 is increasing or decre… - Vidyadip

Question

Find the interval for which $f(x)=x^4-2 x^2$ is increasing or decreasing.

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Answer

Given $\quad f(x)=x^4-2 x^2$
So,$f^{\prime}(x)=4 x^3-4 x$
$\begin{array}{l}f^{\prime}(x)=4 x\left(x^2-1\right) \\ f^{\prime}(x)=4 x(x+1)(x-1)\end{array}$
For $f(x)$ is increasing $f^{\prime}(x) > 0$ $ \begin{array}{lr} \text { So, } 4 x(x+1)(x-1)>0 \\ \Rightarrow & x(x+1)(x-1) > 0 \\ \Rightarrow & -1< x <0 \text { or } x>1 \\ \Rightarrow & x \in(-1,0) \cup(1, \infty) \end{array} $

So, $f(x)$ is increasing in interval $(-1,0) \cup(1, \infty)$.
$\begin{array}{lc}f(x) & \text { is decreasing } f^{\prime}(x)<0 \\ \Rightarrow & 4 x(x+1)(x-1)< 0 \\ \Rightarrow & x(x+1)(x-1)< 0 \quad \therefore 4 > 0 \\ \Rightarrow & x< -1 \text { or } 0< x <1 \\ \Rightarrow & x \in(-\infty,-1) \cup(0,1)\end{array}$

So, $f(x)$ is decreasing in interval $(-\infty,-1) \cup(0,1)$

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