Is #f(x)=(x^2-6x-12)/(x+2)# increasing or decreasing at #x=1#?

Answer 1

Ezra Adams

Increasing (with slope #5/9#)

#f(x) = (x^2-6x-12)/(x+2) = ((x^2+2x)-(8x+16)+4)/(x+2) = x-8+4/(x+2)#

So:

#f'(x) = 1-4/(x+2)^2#

and:

#f'(1) = 1 - 4/(1+2)^2 = 1 - 4/9 = 5/9 > 0#

So #f(x)# is increasing at #x=1#

graph{(x^2-6x-12)/(x+2) [-8.59, 11.41, -7.28, 2.72]}

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Answer 2

Aurora Alvarado

To determine whether ( f(x) = \frac{x^2 - 6x - 12}{x + 2} ) is increasing or decreasing at ( x = 1 ), we evaluate the derivative of ( f(x) ) at ( x = 1 ). If the derivative is positive, the function is increasing at that point; if it's negative, the function is decreasing.

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

[ f'(x) = \frac{(x + 2)(2x - 6) - (x^2 - 6x - 12)(1)}{(x + 2)^2} ]

Evaluate ( f'(1) ) to determine the behavior at ( x = 1 ). If ( f'(1) > 0 ), then ( f(x) ) is increasing at ( x = 1 ); if ( f'(1) < 0 ), then ( f(x) ) is decreasing 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.