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

Question

Charles Anctil · Accepted Answer

#" "#
Please read the 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:

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

Aurora Alvarado · Answer

undefined Critical points on a graph can be identified by examining where the derivative of the function is zero or undefined. These points include local maximums, local minimums, and points of inflection. To find critical points, locate where the derivative changes sign or is equal to zero. Then, analyze the behavior of the function around those points to determine their nature.