Is #f(x)=(-x^2+3x+2)/(x^2-1)# increasing or decreasing at #x=2#?
graph{(-x^2+3x+2)/(x^2-1) [-8.62, 11.38, -4.16, 5.84]}
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To determine whether the function ( f(x) = \frac{-x^2 + 3x + 2}{x^2 - 1} ) is increasing or decreasing at ( x = 2 ), we can examine the sign of the derivative at that point.
First, find the derivative of ( f(x) ) with respect to ( x ), then evaluate it at ( x = 2 ). If the derivative is positive, the function is increasing at that point. If it is negative, the function is decreasing.
Let's find the derivative ( f'(x) ) using the quotient rule:
[ f'(x) = \frac{(x^2 - 1)(-2x) - (-x^2 + 3x + 2)(2x)}{(x^2 - 1)^2} ]
[ = \frac{-2x^3 + 2x + 2x^3 - 6x^2 - 4x}{(x^2 - 1)^2} ]
[ = \frac{-6x^2 - 4x}{(x^2 - 1)^2} ]
Now, evaluate ( f'(x) ) at ( x = 2 ):
[ f'(2) = \frac{-6(2)^2 - 4(2)}{(2^2 - 1)^2} ]
[ = \frac{-24 - 8}{(4 - 1)^2} ]
[ = \frac{-32}{9} ]
Since ( f'(2) ) is negative, the function ( f(x) ) is decreasing at ( x = 2 ).
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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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