Extrema

Extrema on an Interval

By Sahel Otoom

Multiple Choice

What is an Absolute Maximum / Minimum

• The highest/lowest point for the entire function/graph.

• Any peak in the function

• The part that tends to infinity or negative infinity of the graph

Multiple Choice

f(x) = x² + 1 has both a minimum and a maximum on the closed interval [–1, 2].

• True

• False

Multiple Choice

What is the absolute minimum

of this function?

• None since

  $$y\rightarrow\infty$$

• y=0

  
  

• y=4

• x=4

Multiple Choice

What is the location of the absolute maximum for the graphed function?

• x = 1

• x = 3

• x = 4

• x = 5

Multiple Choice

How many relative extrema are in the picture? (Remember that Absolute extrema are also relative)

• 2

• 3

• 4

Multiple Choice

What is the minimum value of the function graphed?

• (1,-1)

• (3,1)

• y = -1

Multiple Choice

The blue dot on this graph represents a(n)...

• Absolute Maximum

• Relative Maximum

• Relative Minimum

Open Ended

Find the critical numbers of

$$f\left(x\right)=x^3-3x\$$  on the interval on [-2,3]

Multiple Select

Find all the critical numbers of

f(x) = 2x³ – 6x² + 12 on the interval on [-2,3]

• x= 4

• x= 0 and x=2

• x= -2 and x=9

• x= 2

Multiple Select

$$f\left(x\right)=\frac{2}{3}x^3-6x^2+16x+14$$   Find the critical numbers of f(x) on the interval on [1,8]

• x=2 and x=4

• x=0 and x=8

• x = 40

Open Ended

Find the critical numbers of

$$f\left(x\right)=x^4-4x^3-8x^2+1$$  on the interval [-1,5]

Multiple Select

Find all the critical numbers of

f(x) = 6x⁴ – 16x³ – 180x² + 456.

on the interval on [-4,1]

• x= -3 , 0 and 5

• x= -5 ,3 and 9

• x= 10

• x = 7

Open Ended

Find all the critical numbers of

$$f\left(x\right)=\frac{2x^2}{x+2}$$  

Multiple Choice

Find the critical number of f(x) = 4x² – 8x + 1

• x = 2

• x = 0

• x = 8

• x = 1

Multiple Select

$$f\left(x\right)=\frac{1}{3}x^3+3x^2+8x-5$$   Find the critical numbers of f(x).   Select ALL that apply

• x=2

• x=0

• x= -4

• x= -2

Multiple Select

Find all the critical numbers of f(x).

$$f\left(x\right)=\frac{1}{4}x^4-\frac{5}{3}x^3-7x^2+13$$   Select ALL that apply.

• x= 7

• x= 5

• x= 0

• x = -2

• x = 14

Multiple Select

Given  $$0\le x\le2\pi$$  , Find ALL the critical numbers in 

$$f\left(x\right)=\cos x$$  . Select ALL that apply.

• $$x=0$$  

• $$x=\pi$$  

• $$x=2\pi$$  

• $$x=\frac{3\pi}{2}$$  

• $$x=\frac{\pi}{2}$$  

Multiple Choice

Find the critical number of

$$f\left(x\right)=2e^x-2x$$  

• x = 2

• x = 0

• x = 1

• x = e

Open Ended

Find the extrema of f(x) = 3x⁴ – 4x³ on the interval [–1, 2].

Multiple Choice

Find the extrema of f(x) = 3x³ – 3x + 1 on the interval [–2, 2].

• Absolute Max at

  (2 , 3) and Absolute Min at

  (-1 , -3)

• Absolute Max at

  (-2 , 3) and Absolute Min at

  (2 , -3)

• Absolute Max at

  (-2 , -1) and Absolute Min at

  (-1 , -1)

Poll

Choose the picture that describe your feeling about today's lesson