Extreme Min Max of Polynomials

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

A polynomial of degree n has at most ​ (a)   zeros

and at most ​ (b)   extrema.

• n

• n-1

• n+1

• 2n

• n^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

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 this graphed polynomial...

• The domain

  is (-∞, ∞)

• The range

  is (-∞, ∞)

• The range

  is [-4, ∞)

• There is one absolute minimum

• There are no local maximum

Answer explanation

The domain of a polynomial is always (-∞, ∞). The range is [-4, ∞) indicating a minimum value of -4. There is one absolute minimum at this point, and no local maximum exists in this polynomial.

Tags

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

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

What is the predicted end behavior of: y=-2x⁴+3x-1

• up on both ends

• down on both ends

• up on left, down on right

• down on left, up on right

Answer explanation

The leading term is -2x^4, which is negative. As x approaches ±∞, the function y approaches -∞. Therefore, the end behavior is down on both ends.

Tags

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

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

What are the extrema of the graph?

• Max = (2, -3), (1, 2)

  Min = (-3, -1), (-1, 4)

• Max = (-3, -1), (-1, 4)

  Min = (2, -3), (1, 2)

• Max = (-1, -3), (4, -1)

  Min = (-3, 2), (2, 1)

• Max = (-3, 2), (2, 1)

  Min = (-1, -3), (4, -1)

What are the extrema of the graph?

• Max = (0,-4)

  Min = (-1.5,0) and (1.5, 0)

• Max = (0,1.5) and (0, -1.5)

  Min = (-4,0)

• Max = (-1.5,0) and (1.5, 0)

  Min = (0,-4)

• Max = (-4,0)

  Min =(0,1.5) and (0, -1.5)

What are the increasing and decreasing intervals of the graph?

• Increasing = (-1.5, 0) & (1.5, ∞)

  Decreasing = (-∞, -1.5) & (0, 1.5)

• Increasing = (-∞, -1.5) & (0, 1.5)

  Decreasing = (-1.5, 0) & (1.5, ∞)

• Increasing = (-∞, -1.5) & (1.5, ∞)

  Decreasing = (-1.5, 0) & (0, 1.5)

• Increasing = (-1.5, 0) & (0, 1.5)

  Decreasing = (-∞, -1.5) & (1.5, ∞)

What are the decreasing intervals on the graph?

• (-∞, 0) & (-4, ∞)

• (-∞, -1) & (1, ∞)

• (-∞, -2) & (2, ∞)

• (-∞, 0) & (4, ∞)