GRAPHS ON THE GO: Understanding polynomials - (Monotonicity)

A very important concept to understand graphs and their nature which leads to better graph plotting.

To understand any polynomial and trace its graph 3 factors are highly important-

1. Roots

2. Monotonicity

3. Concavity

1.Roots- We already know about them, they are the values of x for which the function attains 0.

2. Monotonicity- This tells us the nature of polynomial for different intervals.

It helps us in understanding where the graph increases , decreases of remains constant.

This can be determined by determining the rate of change of function wrt  x which is given by finding the first derivative of the function.

The expression obtained by first derivative is put equal to zero and critical points are found.

After finding critical points they are plotted the marked on a line in ascending order and then different results are drawn according to the results obtained-

if first derivative is(for the particular interval)-

1. Positive                 then the function is increasing

2. Negative                then the function is decreasing

3. Zero                       the function is constant.

Now lets understand the same graphically-

The function plotted below is-

y = x³ - x

with roots 0 , 1 , -1

Now if we take its first derivative-

we get-

y = 3x² - 1

and put it equal to zero

we get two points (-1/sqrt(3) and 1/sqrt(3)) at which the tangent to the curve becomes parallel to the x axis, these points are called critical points.

These points are called critical points because after this the graph may change its nature or may become discontineous or may become non differentiable.

At these points generally function changes its nature as obvious by above graph.

Let's check it monotonicity according to graph-

1. for     (-inf , -1/sqrt(3) )               -           increasing

2. for (-1/sqrt(3) , 1/sqrt(3))            -           decreasing

3. for (1/sqrt(3) , inf )                      -           increasing

Now doing the same mathematically-

Hope the concept of monotonicity is clear.

In next post we will learn about the concavity.

(Image credits- https://www.desmos.com/)

Stay Tuned for the next post.

For any query comment down there.

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