Is #f(x)=(12x^2-16x-12)/(x+2)# increasing or decreasing at #x=5#?

Answer 1

#"increasing at "x=5#

#"to determine if f(x) is increasing at x = a evaluate"# #f'(x)" at x = a"#
#• " if "f'(a)>0" then f(x) is increasing at x = a"#
#• " if "f'(a)<0" then f(x) is decreasing at x = a"#
#"differentiate using the "color(blue)"quotient rule"#
#"given "f(x)=(g(x))/(h(x))" then"#
#f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larr" quotient rule"#
#g(x)=12x^2-16x-12rArrg'(x)=24x-16#
#h(x)=x+2rArrh'(x)=1#
#rArrf'(x)=((x+2)(24x-16)-(12x^2-16x-12))/(x+2)^2#
#rArrf'(5)=(728-208)/49>0#
#rArrf(x)" is increasing at x = 5"#
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Answer 2

To determine whether ( f(x) = \frac{{12x^2 - 16x - 12}}{{x + 2}} ) is increasing or decreasing at ( x = 5 ), we can evaluate the derivative of the function at that point.

First, find the derivative of ( f(x) ) using the quotient rule:

[ f'(x) = \frac{{(12x^2 - 16x - 12)'(x + 2) - (12x^2 - 16x - 12)(x + 2)'}}{{(x + 2)^2}} ]

[ f'(x) = \frac{{(24x - 16)(x + 2) - (12x^2 - 16x - 12)(1)}}{{(x + 2)^2}} ]

[ f'(x) = \frac{{24x^2 + 8x - 32x - 32 - 12x^2 + 16x + 12}}{{(x + 2)^2}} ]

[ f'(x) = \frac{{12x^2 - 16}}{{(x + 2)^2}} ]

Now, plug in ( x = 5 ) to determine whether the derivative is positive or negative:

[ f'(5) = \frac{{12(5)^2 - 16}}{{(5 + 2)^2}} ]

[ f'(5) = \frac{{12(25) - 16}}{{49}} ]

[ f'(5) = \frac{{288}}{{49}} ]

Since ( f'(5) > 0 ), ( f(x) ) is increasing at ( x = 5 ).

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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.

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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