iven the function f left parenthesis x right parenthesis equals x superscript 4 baseline plus 6 x cubed minus x squared minus 30 x plus 4, use the Intermediate Value Theorem to decide which of the following intervals contains at least one zero. Select all that apply.
(2 points)
Responses
left bracket negative 5 comma negative 4 right bracket
Image with alt text: left bracket negative 5 comma negative 4 right bracket
left bracket negative 4 comma negative 3 right bracket
Image with alt text: left bracket negative 4 comma negative 3 right bracket
left bracket negative 3 comma negative 2 right bracket
Image with alt text: left bracket negative 3 comma negative 2 right bracket
left bracket negative 1 comma 0 right bracket
Image with alt text: left bracket negative 1 comma 0 right bracket
left bracket 0 comma 1 right bracket
Image with alt text: left bracket 0 comma 1 right bracket
left bracket 1 comma 2 right bracket
Image with alt text: left bracket 1 comma 2 right bracket
To determine which intervals contain at least one zero, we can evaluate the function at the endpoints of each interval and check if the signs of the function values change.
1. For the interval [-5, -4]:
f(-5) = (-5)^4 + 6(-5)^3 - (-5)^2 - 30(-5) + 4 = 375 - 750 + 25 + 150 + 4 = -196
f(-4) = (-4)^4 + 6(-4)^3 - (-4)^2 - 30(-4) + 4 = 256 - 384 - 16 + 120 + 4 = -20
Since the endpoints have opposite signs, there is at least one zero in the interval [-5, -4].
2. For the interval [-4, -3]:
f(-4) = -20
f(-3) = (-3)^4 + 6(-3)^3 - (-3)^2 - 30(-3) + 4 = 81 - 162 - 9 + 90 + 4 = 4
Since the endpoints have the same sign, there is no zero in the interval [-4, -3].
3. For the interval [-3, -2]:
f(-3) = 4
f(-2) = (-2)^4 + 6(-2)^3 - (-2)^2 - 30(-2) + 4 = 16 - 48 - 4 + 60 + 4 = 28
Since the endpoints have the same sign, there is no zero in the interval [-3, -2].
4. For the interval [-1, 0]:
f(-1) = (-1)^4 + 6(-1)^3 - (-1)^2 - 30(-1) + 4 = 1 - 6 - 1 + 30 + 4 = 28
f(0) = 0^4 + 6(0)^3 - 0^2 - 30(0) + 4 = 0
Since the endpoints have opposite signs, there is at least one zero in the interval [-1, 0].
5. For the interval [0, 1]:
f(0) = 0
f(1) = (1)^4 + 6(1)^3 - (1)^2 - 30(1) + 4 = 1 + 6 - 1 - 30 + 4 = -20
Since the endpoints have opposite signs, there is at least one zero in the interval [0, 1].
6. For the interval [1, 2]:
f(1) = -20
f(2) = (2)^4 + 6(2)^3 - (2)^2 - 30(2) + 4 = 16 + 48 - 4 - 60 + 4 = 4
Since the endpoints have the same sign, there is no zero in the interval [1, 2].
Based on the Intermediate Value Theorem, the intervals [-5, -4], [-1, 0], and [0, 1] contain at least one zero.