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

The only hypothesis required by the mean value theorem is that

As:

We can calculate the definite integral as:

#int_1^3 (9-x^2)/(4x)dx = int_1^3 (9/(4x)-x/4)dx=[9/4lnx-x^2/8]_1^3= 9/4ln3-9/8+1/8=9/4ln3-1#

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I will assume that you are referring to the Mean Value Theorem for Derivatives.

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}}{{4x}} ) satisfies the hypotheses of the Mean Value Theorem on the interval ([1, 3]) and find ( c ), you need to check two conditions:

**Continuity**: Verify if ( f(x) ) is continuous on the interval ([1, 3]).**Differentiability**: Confirm if ( f(x) ) is differentiable on the interval ((1, 3)).

Once you've confirmed both conditions, you can find ( c ) by applying the Mean Value Theorem, which states that if a function ( f ) is continuous on ([a, b]) and differentiable on ((a, b)), then there exists a ( c ) in ((a, b)) such that ( f'(c) = \frac{{f(b) - f(a)}}{{b - a}} ).

After verifying the hypotheses of the Mean Value Theorem, to find ( c ), calculate ( f(1) ) and ( f(3) ), then find the derivative ( f'(x) ). Finally, use the formula ( f'(c) = \frac{{f(3) - f(1)}}{{3 - 1}} ) to solve for ( c ).

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