# How do I find the numbers #c# that satisfy Rolle's Theorem for #f(x)=sqrt(x)-x/3# on the interval #[0,9]# ?

Let us look at some details.

By taking the derivative,

by taking the reciprocal,

by dividing by 2,

by squaring,

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To find the numbers ( c ) that satisfy Rolle's Theorem for the function ( f(x) = \sqrt{x} - \frac{x}{3} ) on the interval ([0,9]), we need to check if the conditions of Rolle's Theorem are met:

- ( f(x) ) is continuous on the closed interval ([0,9]).
- ( f(x) ) is differentiable on the open interval ((0,9)).
- ( f(0) = f(9) ).

Let's evaluate these conditions:

- ( f(x) = \sqrt{x} - \frac{x}{3} ) is continuous on the closed interval ([0,9]) because it is a combination of continuous functions.
- ( f(x) = \sqrt{x} - \frac{x}{3} ) is differentiable on the open interval ((0,9)) because both ( \sqrt{x} ) and ( \frac{x}{3} ) are differentiable on that interval.
- ( f(0) = \sqrt{0} - \frac{0}{3} = 0 ) and ( f(9) = \sqrt{9} - \frac{9}{3} = 3 - 3 = 0 ). Therefore, ( f(0) = f(9) ).

Since all conditions of Rolle's Theorem are satisfied, there exists at least one number ( c ) in the open interval ((0,9)) such that ( f'(c) = 0 ).

To find ( c ), we need to find the derivative of ( f(x) ) and set it equal to zero, then solve for ( x ).

( f'(x) = \frac{1}{2\sqrt{x}} - \frac{1}{3} )

Setting ( f'(x) ) equal to zero:

( \frac{1}{2\sqrt{x}} - \frac{1}{3} = 0 )

( \frac{1}{2\sqrt{x}} = \frac{1}{3} )

Solving for ( x ):

( \sqrt{x} = \frac{3}{2} )

( x = \left(\frac{3}{2}\right)^2 )

( x = \frac{9}{4} )

So, ( c = \frac{9}{4} ). Therefore, the number ( c ) that satisfies Rolle's Theorem for the given function on the interval ([0,9]) is ( \frac{9}{4} ).

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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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