To apply the Intermediate Value Theorem (IVT) to show that the polynomial \( f(x) = 2x^4 - 10x^2 + 2 \) has a real zero between -2 and 0, we need to evaluate the function at the endpoints of the interval and check the sign of the function values.
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Evaluate \( f(-2) \): \[ f(-2) = 2(-2)^4 - 10(-2)^2 + 2 \] \[ = 2(16) - 10(4) + 2 \] \[ = 32 - 40 + 2 = -6 \]
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Evaluate \( f(0) \): \[ f(0) = 2(0)^4 - 10(0)^2 + 2 \] \[ = 0 - 0 + 2 = 2 \]
Now we have:
- \( f(-2) = -6 \) (which is less than 0)
- \( f(0) = 2 \) (which is greater than 0)
Since \( f(-2) < 0 \) and \( f(0) > 0 \), by the Intermediate Value Theorem, there exists at least one real zero of \( f(x) \) in the interval \( (-2, 0) \).
Using the provided options, the correct choice would be:
O B. Because f(x) is a polynomial with \( f(-2) = -6 < 0 \) and \( f(0) = 2 > 0 \), the function has a real zero between -2 and 0.
Fill in:
- \( f(-2) = -6 \) and \( f(0) = 2 \)
- the function has a real zero between -2 and 0.