About The Intermediate Value Theorem:
The intermediate value theorem tells us: if f(a) and f(b) are different signs, then there must be at least one real zero between a and b. For the upper and lower bounds, we can use synthetic division to test potential values and determine if they are an upper bound, meaning there won't be a real zero above or a lower bound, meaning there won't be a real zero below.
Test Objectives
- Demonstrate the ability to use the intermediate value theorem
- Demonstrate the ability to find an upper bound
- Demonstrate the ability to find a lower bound
#1:
Instructions: Show there is a zero between the two numbers given.
a) f(x) = x5 - 3x4 + 4x3 - 12x2 - 12x + 36
-1, 2
b) f(x) = x3 + 7x2 + 13x + 3
-4, 0
Watch the Step by Step Video Lesson View the Written Solution
#2:
Instructions: Determine if k is an upper bound, a lower bound, or neither.
a) f(x) = 2x4 + 10x3 + 14x + 6x
k = 2
b) f(x) = x5 - x3 - 2x2 + 5x - 3
k = -5
Watch the Step by Step Video Lesson View the Written Solution
#3:
Instructions: Determine if k is an upper bound, a lower bound, or neither.
a) f(x) = 4x5 - 16x4 - 8x3 + 16x2 + 4x
k = -1
b) f(x) = 2x4 - 2x3 - 5x2 - 3x + 2
k = 3
Watch the Step by Step Video Lesson View the Written Solution
#4:
Instructions: Determine if k is an upper bound, a lower bound, or neither.
a) f(x) = x4 + 7x3 - 25x2 + 11x + 6
k = 1
b) f(x) = 4x5 - 11x4 - 18x3 - 7x2 - 4x
k = -4
Watch the Step by Step Video Lesson View the Written Solution
#5:
Instructions: Determine if k is an upper bound, a lower bound, or neither.
a) f(x) = 3x4 - 14x3 + 11x2 - 14x + 8
k = 1
b) f(x) = 3x4 - x3 - 2x2 + 3x + 2
k = -4
Watch the Step by Step Video Lesson View the Written Solution
Written Solutions:
#1:
Solutions:
a) f(-1) = 28, f(2) = -20, f(-1) > 0, f(2) < 0
b) f(-4) = -1, f(0) = 3, f(-4) < 0, f(0) > 0
Watch the Step by Step Video Lesson
#2:
Solutions:
a) upper bound
b) lower bound
Watch the Step by Step Video Lesson
#3:
Solutions:
a) neither
b) upper bound
Watch the Step by Step Video Lesson
#4:
Solutions:
a) neither
b) lower bound
Watch the Step by Step Video Lesson
#5:
Solutions:
a) neither
b) lower bound
