# Is #f(x)=(2x-2)(x+1)(x+4)# increasing or decreasing at #x=1#?

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To determine if the function ( f(x) = (2x - 2)(x + 1)(x + 4) ) is increasing or decreasing at ( x = 1 ), we can analyze the sign of the derivative at that point.

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

[ f'(x) = (2)(x + 1)(x + 4) + (2x - 2)(1)(x + 4) + (2x - 2)(x + 1)(1) ]

Now, evaluate ( f'(1) ):

[ f'(1) = (2)(1 + 1)(1 + 4) + (2(1) - 2)(1)(1 + 4) + (2(1) - 2)(1 + 1)(1) ] [ f'(1) = (2)(2)(5) + (0)(5) + (0)(2) ] [ f'(1) = 20 ]

Since ( f'(1) = 20 > 0 ), the function is increasing at ( x = 1 ).

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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.

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- What are the critical points of #f(x,y) = x^3 + y^3 - xy#?

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