# How do you determine all values of c that satisfy the mean value theorem on the interval [1, 4] for #f(x)=1/sqrt(x)#?

To determine all values of ( c ) that satisfy the Mean Value Theorem (MVT) on the interval ([1, 4]) for ( f(x) = \frac{1}{\sqrt{x}} ), you need to find the average rate of change of the function over that interval and then find where this average rate of change equals the instantaneous rate of change.

The average rate of change of ( f(x) ) over the interval ([a, b]) is given by:

[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} ]

For ( f(x) = \frac{1}{\sqrt{x}} ) on the interval ([1, 4]), we have:

[ f(1) = \frac{1}{\sqrt{1}} = 1 ] [ f(4) = \frac{1}{\sqrt{4}} = \frac{1}{2} ]

[ \text{Average rate of change} = \frac{\frac{1}{2} - 1}{4 - 1} = -\frac{1}{6} ]

To apply the Mean Value Theorem, find the derivative of ( f(x) ) and set it equal to the average rate of change:

[ f'(x) = \frac{-1}{2x^{3/2}} ]

Setting this equal to ( -\frac{1}{6} ) and solving for ( x ) gives:

[ \frac{-1}{2x^{3/2}} = -\frac{1}{6} ] [ x^{3/2} = 3 ] [ x = 3^{\frac{2}{3}} ]

Thus, the value of ( c ) that satisfies the Mean Value Theorem on the interval ([1, 4]) for ( f(x) = \frac{1}{\sqrt{x}} ) is ( c = 3^{\frac{2}{3}} ).

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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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