How do you use the mean value theorem to explain why (-1) is the only zero of the function f(x)=x³+3x²+9x+7?
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To show that (-1) is the only zero of the function f(x) = x³ + 3x² + 9x + 7 using the Mean Value Theorem, we first need to show that f(x) has exactly one real root. We can do this by showing that f(x) is continuous and changes sign at x = -1.
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Show continuity of f(x): The function f(x) = x³ + 3x² + 9x + 7 is a polynomial, so it is continuous for all real numbers x.
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Show that f(-1) > 0 and f(0) < 0: Calculate f(-1) and f(0) to determine the sign change around x = -1.
f(-1) = (-1)³ + 3(-1)² + 9(-1) + 7 = -1 + 3 - 9 + 7 = 0 f(0) = 0³ + 3(0)² + 9(0) + 7 = 7
Since f(-1) = 0 and f(0) = 7, f(x) changes sign at x = -1.
- Apply the Mean Value Theorem: Since f(x) is continuous on the interval [-1, 0] and differentiable on (-1, 0), the Mean Value Theorem guarantees the existence of a c in (-1, 0) such that:
f'(c) = (f(0) - f(-1)) / (0 - (-1)) = (7 - 0) / (0 + 1) = 7
However, f'(x) = 3x² + 6x + 9, and this derivative is always positive for all real x. Therefore, there cannot be any c in (-1, 0) where f'(c) = 7. This contradiction shows that f(x) has no zero in (-1, 0), and since we know it has a zero at x = -1, this must be the only zero of the function.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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