Maxima and Minima of Graphs of Polynomials

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

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

What is the relative maximum?

• (2,0)

• (4.67, -9.48)

What is the relative maximum?

• (5, -4)

• (3, 0)

What is the relative max?

• (0, 36)

• (-2.55, -6.55), (2.55, -6.55)

• (-∞, ∞)

• None

Which of the following statement is FALSE?

• There is a relative min at the point (8, 0)

• There is another relative min at the point (0, -7.5)

• There is a relative max at the point (5, 2.5)

• There are more relative maximums than there are relative minimums.

Answer explanation

The statement 'There are more relative maximums than there are relative minimums' is false because the given points indicate there are two relative minima and only one relative maximum.

Tags

CCSS.MATH.CONTENT.HSF.IF.B.4

CCSS.MATH.CONTENT.HSF.IF.C.7.c

Select all statements that apply to the graphed polynomial...

• The domain

  is (-∞, ∞)

• The range

  is (-∞, ∞)

• The range

  is (-∞, 6)

• There is 1 absolute maximum

• There are 1 local minimum, 1 local maximum

Answer explanation

The domain of a polynomial is always (-∞, ∞). The range is also (-∞, ∞) if it does not have a maximum or minimum. The graph shows 1 local minimum and 1 local maximum, confirming these statements.

Tags

CCSS.MATH.CONTENT.HSF.IF.B.4

CCSS.MATH.CONTENT.HSF.IF.C.7.c

• 0

• 1

• 4

• -1

• 2

• 5

Answer explanation

To find extrema, we calculate the derivative f'(x) = -6x^2 + 24x - 8 and set it to zero. Solving gives x = 0, 1, and 4. The relative minimum occurs between x = 0 and x = 1, and the relative maximum is at x = 4.

Tags

CCSS.MATH.CONTENT.HSF.IF.B.4

CCSS.MATH.CONTENT.HSF.IF.C.7.c

Estimate the x-coordinate at which extrema occur.

• 0

• 1

• -1

• 2

• 5

• -2

• 3

• 4

• -3

Answer explanation

To find extrema, we calculate the derivative f'(x) = 6x^2 - 8x - 3 and set it to zero. Solving gives critical points at x = 2 and x = -1. The relative minimum occurs between x = -1 and x = 0, and the relative maximum between x = 1 and x = 2.

Tags

CCSS.MATH.CONTENT.HSF.IF.B.4

CCSS.MATH.CONTENT.HSF.IF.C.7.c

f(x) = 2x³ - 3x² + 4x -10
State the maximum number of turns the graph of f(x) could make.

• 3

• 2

• 1

• 0

Tags

CCSS.MATH.CONTENT.HSF.IF.B.4

CCSS.MATH.CONTENT.HSF.IF.C.7.c