To apply the Intermediate Value Theorem (IVT), we need to evaluate the function \( f(x) = x^4 + 6x^3 - x^2 - 30x + 4 \) at specific points within the provided intervals. The IVT states that if \( f(a) \) and \( f(b) \) have opposite signs, then there is at least one zero of the function in the interval \( (a, b) \).
Let's evaluate the function at the endpoints of each interval:
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Interval \([-5, -4]\):
- \( f(-5) = (-5)^4 + 6(-5)^3 - (-5)^2 - 30(-5) + 4 = 625 - 750 - 25 + 150 + 4 = 4 \)
- \( f(-4) = (-4)^4 + 6(-4)^3 - (-4)^2 - 30(-4) + 4 = 256 - 384 - 16 + 120 + 4 = -20 \)
- Since \( f(-5) = 4 > 0 \) and \( f(-4) = -20 < 0 \), the IVT applies. There is at least one zero in \([-5, -4]\).
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Interval \([-4, -3]\):
- \( f(-4) = -20 \) (as calculated above).
- \( f(-3) = (-3)^4 + 6(-3)^3 - (-3)^2 - 30(-3) + 4 = 81 - 162 - 9 + 90 + 4 = 4 \)
- Since \( f(-4) = -20 < 0 \) and \( f(-3) = 4 > 0 \), the IVT applies. There is at least one zero in \([-4, -3]\).
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Interval \([-3, -2]\):
- \( f(-3) = 4 \) (as calculated above).
- \( f(-2) = (-2)^4 + 6(-2)^3 - (-2)^2 - 30(-2) + 4 = 16 - 48 - 4 + 60 + 4 = 28 \)
- Since both \( f(-3) = 4 > 0 \) and \( f(-2) = 28 > 0 \), the IVT does not apply here. There is no zero in \([-3, -2]\).
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Interval \([-1, 0]\):
- \( f(-1) = (-1)^4 + 6(-1)^3 - (-1)^2 - 30(-1) + 4 = 1 - 6 - 1 + 30 + 4 = 28 \)
- \( f(0) = 0^4 + 6(0)^3 - 0^2 - 30(0) + 4 = 4 \)
- Since both \( f(-1) = 28 > 0 \) and \( f(0) = 4 > 0 \), the IVT does not apply here. There is no zero in \([-1, 0]\).
-
Interval \([0, 1]\):
- \( f(0) = 4 \) (as calculated above).
- \( f(1) = (1)^4 + 6(1)^3 - (1)^2 - 30(1) + 4 = 1 + 6 - 1 - 30 + 4 = -20 \)
- Since \( f(0) = 4 > 0 \) and \( f(1) = -20 < 0 \), the IVT applies. There is at least one zero in \([0, 1]\).
-
Interval \([1, 2]\):
- \( f(1) = -20 \) (as calculated above).
- \( f(2) = (2)^4 + 6(2)^3 - (2)^2 - 30(2) + 4 = 16 + 48 - 4 - 60 + 4 = 4 \)
- Since \( f(1) = -20 < 0 \) and \( f(2) = 4 > 0 \), the IVT applies. There is at least one zero in \([1, 2]\).
Thus, the intervals that contain at least one zero are:
- \([-5, -4]\)
- \([-4, -3]\)
- \([0, 1]\)
- \([1, 2]\)
In conclusion, the intervals containing at least one zero are:
- \([-5, -4]\)
- \([-4, -3]\)
- \([0, 1]\)
- \([1, 2]\)