Question
Find the interval for which $f(x)=x^4-2 x^2$ is increasing or decreasing.
✓
Answer
Given $ f(x)=x^4-2 x^2$
So,$f^{\prime}(x)=4 x^3-4 x$
$f^{\prime}(x)=4 x\left(x^2-1\right)$
$f^{\prime}(x)=4 x(x+1)(x-1)$
For $f(x)$ is increasing $f^{\prime}(x) > 0$
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) $
So, $f(x)$ is increasing in interval $(-1,0) \cup(1, \infty)$.
${lc}f(x)$ is decreasing $f^{\prime}(x)<0$
$\Rightarrow 4 x(x+1)(x-1)< 0$
$\Rightarrow x(x+1)(x-1)< 0 \therefore 4 > 0$
$\Rightarrow x< -1 \text { or } 0< x <1$
$\Rightarrow x \in(-\infty,-1) \cup(0,1)$
So, $f(x)$ is decreasing in interval $(-\infty,-1) \cup(0,1)$
So,$f^{\prime}(x)=4 x^3-4 x$
$f^{\prime}(x)=4 x\left(x^2-1\right)$
$f^{\prime}(x)=4 x(x+1)(x-1)$
For $f(x)$ is increasing $f^{\prime}(x) > 0$
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) $
So, $f(x)$ is increasing in interval $(-1,0) \cup(1, \infty)$.
${lc}f(x)$ is decreasing $f^{\prime}(x)<0$
$\Rightarrow 4 x(x+1)(x-1)< 0$
$\Rightarrow x(x+1)(x-1)< 0 \therefore 4 > 0$
$\Rightarrow x< -1 \text { or } 0< x <1$
$\Rightarrow x \in(-\infty,-1) \cup(0,1)$
So, $f(x)$ is decreasing in interval $(-\infty,-1) \cup(0,1)$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If the position vectors of the points A(3, 4), B(5, -6) and C(4, -1) are $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ respectively, compute $\vec{\text{a}}+2\vec{\text{b}}-3\vec{\text{c}}$.
→If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?
Evaluate the following integrals:
$\int\sqrt{4\text{x}^2-5}\text{dx}$
→$\int\sqrt{4\text{x}^2-5}\text{dx}$
If $\Big(\sin^{-1}\text{x}\Big)^2+\Big(\cos^{-1}\text{x}\Big)^2=\frac{175\pi^2}{36},$ find x.
→If f(x) = |x - 2| write whether f(2) exists or not.
Find the principal values of the following:
$\tan^{-1}\Big(-\frac{1}{\sqrt3}\Big)$
→$\tan^{-1}\Big(-\frac{1}{\sqrt3}\Big)$
If $\text{y}=\text{e}^\text{x}(\sin\text{x}+\cos\text{x})$ prove that $\frac{\text{d}^2\text{y}}{\text{dx}^2}-2\frac{\text{dy}}{\text{dx}}+2\text{y}=0$
→Show that the points whose position vectors are as given below are collinear:
$3\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}},\ \hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $-\hat{\text{i}}+4\hat{\text{j}}-2\hat{\text{k}}$
→$3\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}},\ \hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $-\hat{\text{i}}+4\hat{\text{j}}-2\hat{\text{k}}$
A man wins a rupee for head and loses a rupee for tail when a coin is tossed. Suppose that he tosses once and quits if he wins but tries once more if he loses on the first toss. Find the probability distribution of the number of rupees the man wins.
Let f: R → R be the Signum Function defined as $f(\text{x})=\begin{cases}1,&\text{x}>0\\0,&\text{x}=0\\-1,&\text{x}<0\end{cases}$ and g: R → R be the Greatest Integer Function given by g(x) = [x], where [x] is greatest integer less than or equal to x. Then, does fog and gof coincide in (0, 1]?
→