# How can you identify critical points by looking at a graph?

Aug 7, 2018

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#### Explanation:

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Definition of a Critical Point:

A continuous function color(red)(f(x) has a critical point at that point color(red)(x if it satisfies one of the following conditions:

1. color(blue)(f'(x)=0

2. color(blue)(f'(x) is undefined.

A critical point can be a local maximum if the functions changes from increasing to decreasing at that point OR

a local minimum if the function changes from decreasing to increasing at that point.

color(green)("Example 1:"

Let us consider the Sin Graph:

One Period of this graph is from color(blue)(0 " to " 2pi.

The graph does not go above color(red)((+1) and does not go down below color(red)((-1)

View the graph below:

Note that the graph starts from color(red)(0 and goes up to color(red)(pi/2 then comes down to reach the x-intercept at color(red)(pi, then goes down to minimum at $\left(\textcolor{red}{- \frac{3 \pi}{2}}\right)$ and goes up again to the x-intercept at color(red)(2pi to complete one complete period.

Observe that the points color(blue)(C1, C3 and C5 are the x-intercepts.

We have a maximum at the point color(blue)(C2.

Critical Points:

Formula : color(red)("Period" / B

Note that the distance between the points:

color(green)(0 " to " pi/2

color(green)(pi/2 " to " pi

color(green)(pi " to " (3pi)/2

color(green)((3pi)/2 " to " 2pi

are all equal and there are four of them.

Hence, $B = 4$

rArr color(red)("Period" / 4

$\Rightarrow \frac{2 \pi}{4}$

and the Critical Points are color(blue)(C1, C2, C3, C4 and C5

and the distance between any two critical point is $\frac{\pi}{2}$

Hope this helps.