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

1 Answer
Aug 7, 2018

Answer:

#" "#
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:

enter image source here

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:

enter image source here

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.

enter image source here

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.