Is #f(x)=(x^2-4x-9)/(x+1)# increasing or decreasing at #x=0#?
Increasing
graph{-x^2-4x-9)/(x+1 [-27.72, 18.52, -44.5, 47.97]}
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To determine if ( f(x) = \frac{{x^2 - 4x - 9}}{{x + 1}} ) is increasing or decreasing at ( x = 0 ), we can examine the sign of the derivative of ( f(x) ) at that point.
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Find the derivative of ( f(x) ) using the quotient rule: [ f'(x) = \frac{{(x + 1)(2x - 4) - (x^2 - 4x - 9)(1)}}{{(x + 1)^2}} ]
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Simplify the expression: [ f'(x) = \frac{{2x^2 - 4x + 2x - 4 - x^2 + 4x + 9}}{{(x + 1)^2}} ] [ f'(x) = \frac{{x^2 - 4}}{{(x + 1)^2}} ]
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Evaluate ( f'(0) ): [ f'(0) = \frac{{0^2 - 4}}{{(0 + 1)^2}} = -4 ]
Since ( f'(0) = -4 ) is negative, the function is decreasing at ( x = 0 ).
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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.
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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