# How do you verify that the function #f(x) = (x)/(x+2)# satisfies the hypotheses of the Mean Value Theorem on the given interval [1,4], then find all numbers c that satisfy the conclusion of the Mean Value Theorem?

As for the derivative, using the ratio formula

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To verify that the function ( f(x) = \frac{x}{x+2} ) satisfies the hypotheses of the Mean Value Theorem on the interval ([1,4]), we first check if the function is continuous on the closed interval ([1,4]) and differentiable on the open interval ((1,4)). The function is continuous and differentiable on the given interval.

Next, we find the derivative of ( f(x) ) to apply the Mean Value Theorem. The derivative is given by:

[ f'(x) = \frac{d}{dx}\left(\frac{x}{x+2}\right) = \frac{2}{(x+2)^2} ]

Now, we check if ( f'(x) ) is continuous on ([1,4]), which it is. Therefore, all conditions of the Mean Value Theorem are satisfied.

To find all numbers ( c ) that satisfy the conclusion of the Mean Value Theorem, we apply the theorem:

[ f'(c) = \frac{f(4) - f(1)}{4 - 1} ] [ f'(c) = \frac{\frac{4}{6} - \frac{1}{3}}{3} ] [ f'(c) = \frac{\frac{2}{3} - \frac{1}{3}}{3} ] [ f'(c) = \frac{1}{9} ]

Now, solve for ( c ):

[ \frac{2}{(c+2)^2} = \frac{1}{9} ] [ 18 = (c+2)^2 ] [ c+2 = \pm \sqrt{18} ] [ c+2 = \pm 3\sqrt{2} ] [ c = -2 \pm 3\sqrt{2} ]

So, the numbers ( c ) that satisfy the conclusion of the Mean Value Theorem are ( c = -2 + 3\sqrt{2} ) and ( c = -2 - 3\sqrt{2} ).

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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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