# Given the function #f(x)=(x^2-9)/(3x)#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [1,4] and find the c?

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Answers

Therefore this function satisfies the hypotheses of the Mean Value Theorem on this interval.

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To determine if the function ( f(x) = \frac{{x^2 - 9}}{{3x}} ) satisfies the hypotheses of the Mean Value Theorem on the interval ([1, 4]) and find ( c ), you need to check two conditions:

**Continuity**: ( f(x) ) must be continuous on the closed interval ([1, 4]).**Differentiability**: ( f(x) ) must be differentiable on the open interval ((1, 4)).

First, check the continuity of ( f(x) ) on ([1, 4]) by verifying that there are no points of discontinuity within this interval.

Next, check the differentiability of ( f(x) ) on ((1, 4)) by verifying that the derivative ( f'(x) ) exists and is continuous on ((1, 4)). Compute ( f'(x) ) by differentiating ( f(x) ) with respect to ( x ).

Once you have confirmed continuity and differentiability, apply the Mean Value Theorem to find ( c ). The theorem states that if ( f(x) ) satisfies the hypotheses, there exists at least one ( c ) in the interval ((1, 4)) such that ( f'(c) = \frac{{f(4) - f(1)}}{{4 - 1}} ).

Calculate ( f(4) ) and ( f(1) ), then find ( f'(c) ). Finally, solve for ( c ).

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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.

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