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This is the second of our many quizzes on Increasing and decreasing intervals. Click on this to access more such competitive quizzes on Calculus.

Increasing and decreasing intervals Quiz 2

mathematics

 Find the intervals where $f(x)={{x}^{3}}+27{{x}^{2}}+27x+1$ will increase?

 If ${f}'(x)>0$, then $f(x)$ is said to be increasing. So, ${f}'(x)$ of the function $f(x)={{x}^{3}}+27{{x}^{2}}+27x+1$ have to be found. Now the required intervals where ${f}'(x)$ is greater than zero should be found.

$f(x)={{x}^{3}}+9{{x}^{2}}+27x+1 \\  \Rightarrow {f}'(x)=3{{\left( x+3 \right)}^{2}} \\ $ So, it is clear that $f(x)={{x}^{3}}+27{{x}^{2}}+27x+1$ is an increasing function for all real values of $x$.

 Which of the following statements are correct if $f(x)$ is a decreasing function? 

 A function $f(x)$ is an increasing function if ${f}'(x) > 0$ and a decreasing function if ${f}'(x) < 0$. The graph of decreasing function goes on decreasing as the value of $x$ increases. 

 A function $f(x)$ is a decreasing function if ${f}'(x) < 0$. $ \\ $ The graph of decreasing function goes on decreasing as the value of $x$ increases.

 Which of the following graphs of ${f}'(x)$ represent the exact graph to the given situation? $f(x)$ is always decreasing.

 If ${f}'(x) > 0$, then $f(x)$ is said to be increasing. If ${f}'(x) < 0$, then $f(x)$ is said to be decreasing.

 The graph in option B represents a decreasing function because ${f}'(x) < 0$ at real values of $x$.

 Which of the following statements is correct regarding the following graph? The following graph represents ${f}'(x)$ $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/ad80cd6c-8e60-4be6-8261-51bdb99de9d7764778792617713272.png"/>

 Find whether the value of ${f}'(x)$ in different intervals and also find out in which interval $f\left( x \right)$ is either increasing or decreasing.

$\text{For } x > 0, \text{ it is clear that }{f}'(x) > 0. \\ \text{So, }f\left( x \right)\text{ is increasing if } x > 0. \\ \text{For } x < 0, \text{ it is clear that } {f}'(x) < 0. \\ \text{So, }f\left( x \right)\text{ is decreasing if } x < 0$.

 Find the intervals of $f(x)={{x}^{3}}+3x+\sin x+1$ where the function will decrease?

 First it is required to ${f}'(x)$. $f(x)$ is said to be decreasing when ${f}'(x) < 0$. Based on this find the required intervals to find the range where the function is decreasing.

 $f(x)={{x}^{3}}+3x+\sin x+1 \\  \Rightarrow {f}'(x)=3{{x}^{2}}+\cos x+3 \\ $ A function is said to be decreasing if ${f}'(x) < 0$. $ \\   \Rightarrow 3{{x}^{2}}+\cos x+3<0 \\  \Rightarrow -3({{x}^{2}}+1)>\cos x \\  \Rightarrow -3({{x}^{2}}+1)>1\text{ (or) }-3({{x}^{2}}+1)>-1 \\  \Rightarrow {{x}^{2}}<\dfrac{-4}{3}\text{ (or) }{{x}^{2}}<\dfrac{-2}{3} \\ $ So, it is clear that $f(x)={{x}^{3}}+3x+\sin x+1$ is always an increasing function.