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Is #f(x)=-x/(x-5)# increasing or decreasing at #x=-1#?

Answer 1

Emilia Aguilar

#f(x)=-(x)/(x-5)# is always increasing.

#f(x)=-(x)/(x-5)=-(x-5+5)/(x-5)=-1-5*1/(x-5)#

hence #f'(x)=-5*((-1)/(x-5)^2)=5/(x-5)^2#

As #f'(x)>=0# for all #x#

#f(x)=-(x)/(x-5)# is always increasing.

The graph{-(x)/(x-5 [-10, 10, -5, 5]}

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Answer 2

Cameron Alexander

To determine if ( f(x) = -\frac{x}{x-5} ) is increasing or decreasing at ( x = -1 ), we can analyze the derivative of the function at that point. The derivative of ( f(x) ) with respect to ( x ) is given by:

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

Now, substitute ( x = -1 ) into ( f'(x) ) to find the derivative at ( x = -1 ):

[ f'(-1) = \frac{5}{(-1-5)^2} = \frac{5}{36} ]

Since the derivative ( f'(-1) ) is positive (specifically ( \frac{5}{36} )), it means that the function ( f(x) = -\frac{x}{x-5} ) is increasing at ( x = -1 ).

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Answer from HIX Tutor

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.