Question

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

Answer 1

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Please read the explanation.

Explanation:

`" "`
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 `(color(red)(-(3pi)/2))` 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`

`rArr (2pi)/4`

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

and the distance between any two critical point is `pi/2`

Hope this helps.