Extrema

Extrema

What is Extrema?

Extremas are your maximum and minimum points for example your absolute maximum and minimum are the highest and lowest points on the graph. Then, your local maximum and minimums which are relatives and not the highest or lowest points. Critical points are when the derivative is 0 or undefined.

​Remember this graph for next question!!!

Which point is the absolute minimum? (look at the coordinates (x,y) and remember your answer. Look at the LOWEST point)

Multiple Choice

Which point was the absolute minimum from the previous slide?

• (-3,-2)

• (0,0)

• (2,0)

• (4.1)

A

B

C

D

E

Which Point is the absolute maximum? (answer next slide)

Multiple Choice

Which point was the absolute maximum?

• A

• B

• C

• D

Critical Point

A critical point is a point where the derivative is either 0 or not defined. Critical points are the points on the graph where the function's rate of change is altered by either increasing or decreasing in concavity. Critical point are useful for determining extremas.

​The critical points are the points highlighted because the rate of change has altered from increasing to decreasing at (-1.063, 2.033) then from decreasing to increasing at (0,0).

​REMEMBER YOUR ANSWER FOR NEXT SLIDE!
What are the critical points on this graph? Jot your answers down to answer on the next slide. (Find the (x,y) coordinates, there are 3 answers)

Multiple Select

What were the critical points on the graph?

• (-3,-8)

• (2,0)

• (1,1)

• (4,-3)

How to find critical point of a function?

In order to find the critical points of a function, a derivative has to be taken then set to zero then solve for x.

Multiple Choice

What is the critical point of this function f( x) = 3x⁴ - 4x³?

• (3,5)

• (0,0)

• (1,1)

• (6,2)

Multiple Select

Find the critical points of R(t)= 1+80t³+5t⁴−2t⁵

• 6

• 2

• 0

• -4

Open Ended

How do you find a critical point of a function?

Open Ended

What is an extrema?

How are extrema's used in real life?

In a real life scenario, extrema's can be used for things like economics. One can maximize profits and minimize costs by finding the maximum and minimum points of a function.

Example Problem

​A company produces and sells a certain product. The revenue R(x) is exemplified by selling x units of the product given by the function R(x) = 70x - 2x². Find the number of units the company should.d produce and sell to maximize its revenue.

First you find the derivative of R(x) = 70x - 2x² which is R(x)'= 70 - 4x.. Then we set it equal to 0 so it would be 70 - 4x = 0. Next it would be -4x= -70. We then divide both sides and would get 17.5. Therefore the maximum would be 17.5 units to produce and sell.

Multiple Choice

If a function S(t)= 20x - 2x² is given to represent time, what is the maximum speed one could achieve?

• 3

• 5

• 4

• 6