Given the function #f(x)= abs((x^212)(x^2+4))#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [2,3] and find the c?
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To determine whether the function satisfies the hypotheses of the Mean Value Theorem (MVT) on the interval ([2,3]), you need to check if the function is continuous on the closed interval and differentiable on the open interval.

Continuity: The function (f(x) = \text{abs}((x^2  12)(x^2 + 4))) is continuous everywhere because it is a composition of continuous functions.

Differentiability: To check differentiability, you need to ensure that the function is differentiable on the open interval (2, 3). Differentiability of a function is determined by the differentiability of its components.
(f(x) = \text{abs}((x^2  12)(x^2 + 4))) is differentiable at every point where its components (g(x) = (x^2  12)) and (h(x) = (x^2 + 4)) are differentiable. The functions (g(x)) and (h(x)) are polynomials and hence are differentiable everywhere.
Since the function (f(x)) is continuous on the closed interval [2, 3] and differentiable on the open interval (2, 3), it satisfies the hypotheses of the Mean Value Theorem on the interval [2,3].
To find (c), we use the Mean Value Theorem formula:
[f'(c) = \frac{f(3)  f(2)}{3  (2)}]
Calculate (f(3)) and (f(2)), then find the derivative of (f(x)) and solve for (c).
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When evaluating a onesided 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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