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  • Let a function f be defined as <img alt="f(x)=\left[\begin{array}{ccc} \frac{|x-1|}{x^2+1}, & \text { if } & x>1 \\ x^2, & \text { if } & x \leq-1 \end{array}\righ

Let a function f be defined as

f(x)=\left[\begin{array}{ccc} \frac{|x-1|}{x^2+1}, & \text { if } & x>1 \\ x^2, & \text { if } & x \leq-1 \end{array}\right.

Then, the number of critical point(s) on the graph of this function is/are
 

Option: 1

4


Option: 2

3


Option: 3

2


Option: 4

1


Answers (1)

best_answer


\begin{aligned} & f(x)=\left[\begin{array}{ccc} \frac{x-1}{1+x^2}, & \text { if } x \geq 1 \\ \frac{1-x}{1+x^2}, & \text { if } & -1<x<1 \\ x^2, & \text { if } & x \leq-1 \end{array}\right. \\ & f^{\prime}(x)=0 \text { gives } x=\sqrt{2}+1 \text { or } 1-\sqrt{2} \end{aligned}

The function has a continuity at x=-1.
As x \rightarrow \infty, f(x) \rightarrow 0.
The graph is as shown below


where\frac{d y}{d x}=0, if it exists or \frac{d y}{d x} is non-existent.
The points x=-1,1-\sqrt{2}, 1 and 1+\sqrt{2} are the four critical points on the graph of this function. Hence, (a) is the correct answer.

Posted by

Pankaj Sanodiya

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