# How do you find the critical numbers for #(2-x)/(x+2)^3# to determine the maximum and minimum?

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To find the critical numbers of the function ( f(x) = \frac{2-x}{(x+2)^3} ), we first need to find the derivative of ( f(x) ) with respect to ( x ), then solve for ( x ) when the derivative is equal to zero or undefined.

The derivative of ( f(x) ) can be found using the quotient rule:

[ f'(x) = \frac{d}{dx}\left(\frac{2-x}{(x+2)^3}\right) ]

[ f'(x) = \frac{(x+2)^3(0 - 1) - (2-x)(3(x+2)^2)}{(x+2)^6} ]

[ f'(x) = \frac{- (x+2)^3 + 3(2-x)(x+2)^2}{(x+2)^6} ]

[ f'(x) = \frac{- (x+2)^3 + 3(2x+4)(x+2)^2}{(x+2)^6} ]

[ f'(x) = \frac{- (x+2)^3 + 3(x+2)(2x+4)(x+2)}{(x+2)^6} ]

[ f'(x) = \frac{- (x+2)^3 + 3(x+2)^2(2x+4)}{(x+2)^5} ]

To find the critical numbers, we set ( f'(x) ) equal to zero and solve for ( x ):

[ - (x+2)^3 + 3(x+2)^2(2x+4) = 0 ]

[ - (x+2)^3 + 3(2x+4)(x+2)^2 = 0 ]

After solving this equation, we can determine the values of ( x ) that correspond to the critical numbers of the function. Once we have these critical numbers, we can evaluate ( f(x) ) at these points as well as at the endpoints of any relevant intervals to determine the maximum and minimum values of the function.

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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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