Extrema of Polynomials

A company designs boxes based on their maximum volume. A particular box will have the dimensions for height, length and width as shown the volume equation V(x) = x(20-2x)(8-2x). The domain for x, height in inches is 0<x<4. What is the maximum volume of the crate?

• 1.761 cubic inches

• 129.942 cubic inches

• there is no maximum

• 128 cubic inches

On what interval(s) is this graph decreasing?

A polynomial of degree n will have at most how many extrema?

• n − 1

• n + 1

• n

• n/2

Which one of the following could be the polynomial seen in the image?

• y = (x + 3)(x + 1)(x − 2)

• y = -(x + 3)(x + 1)(x − 2)

• y = -(x − 3)(x + 1)(x + 2)

• y = (x + 3)(x + 1)(x + 2)

Match the EXTREMA

Accordingly

• at x=-2,

  x=3

• Relative

  Minimum(s)

• NONE

• Absolute

  Maximum of

• at x=1

• Relative

  Maximum(s)

• -4

  when x=3

• Absolute

  Minimum of

Match the EXTREMA

Accordingly

• at x=-1

• Absolute

  Maximum of

• NONE

• Relative

  Maximum(s)

• 2

  when x=1

• Relative

  Minimum(s)

• at x=-4,

  x=1

• Absolute

  Minimum of

Choose which given function values will guarantee an extrema.

• f(0)=-2, f(1)=0,

  f(11)=0

• f(-8)=0, f(0)=1,

  f(5)=3

• f(-1)=1 , f(3)=0 ,

  f(4)=0 , f(6)=8

• f(-5)=-8 , f(-3)=-2 ,

  f(0)=0 , f(3)= 7

There are ​ (a)   extrema/turning points. There are ​ (b)   relative maxima. There are ​ (c)   relative minima with multiplicity of 2. There are ​ (d)   absolute extrema because the function is ​ (e)   .

• 4

• 5

• 3

• 1

• 2

• 0

• odd

• even

How many extrema/turning points are there in the function:

• 4

• 3

• 2

• 1

For the quartic function shown, how many and what type of zeros are there?

• 2 real (multiplicity of 2)

  1 real distinct

• 2 real distinct

  1 real (multiplicity of 2)

• 3 real distinct