This is a Test of Mathematics Solution Subjective 124 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

## Problem

** **Sketch on plain paper, the graph of

## Solution

There are several steps to find graph of a function. We will use calculus to analyze the function. Here y=f(x)

**Domain:**The function is defined at all real numbers except x =1 and x = -1 which makes the denominator 0.**Even/Odd:**Clearly f(x) = f(-x). Hence it is sufficient to investigate the function for positive values of x and then reflect it about y axis.**Critical Points:**Next we investigate the critical points.are those values of x for which the first derivative of f(x) is either 0 or undefined. Since , then .**Critical Points**

This implies

Hence critical points are x =0 , 1, -1**Monotonicity:**The first derivative is negative for all positive values of x (note that we are only investigating for positive x values, since we can then reflect the picture about y axis as previously found).*Hence the function is ‘decreasing’ for all positive value of x.***Second Derivative:**We compute the second derivative to understand a couple things:- convexity/concavity of the function
- examine whether the critical points are maxima, minima, inflection points.

Clearly implying at x = 0 we have local maxima. Since f(0) = – 1,**we have (0, -1) as a local maxima**.

Also the second derivative is negative from x = 0 to x = 1 and positive after x = 1**. Hence the curve is under-tangent (concave) from x = 0 to x = 1, and above-tangent (convex) from x =1 onward.**

**Vertical Asymptote:**We next examine what happens*near*x = 1. We want to know what happens when we approach x=1 from*left*and from*right*. To that end we compute the following limits:- (since the denominator gets infinitesimally small with a negative sign, and numerator is about 2)
- (since the denominator gets infinitesimally small with a positive sign, and numerator is about 2)

**Horizontal Asymptote:**Finally we examine what happens when x approaches . To that end we compute the following:

**Drawing the graph:**- Local Maxima at (0,-1)
- Even function hence we draw for positive x values and reflect about y axis
- Vertical asymptote at x =1
- From x = 0 to 1, the function decreasing to negative infinity, staying under tangent all the time.
- From x = 1 to positive infinity, the function decreases from positive infinity to 1 staying above tangent all the time.
- Horizontal Asymptote at y= 1

## Chatuspathi:

**What is this topic:**Graph Sketching using Calculus**What are some of the associated concept:**Maxima, Minima, Derivative, Convexity, Concavity, Asymptotes**Where can learn these topics:**Cheenta**Book Suggestions:**Calculus of One Variable by I.A. Maron, Play with Graph (Arihant Publication)