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

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To determine if ( f(x) = (x - 2)^2(x + 1) ) is increasing or decreasing at ( x = 1 ), we need to evaluate the derivative of ( f(x) ) at that point.

The derivative of ( f(x) ) is given by ( f'(x) ).

( f'(x) = 2(x - 2)(x + 1) + (x - 2)^2 \cdot 1 )

Evaluating ( f'(1) ):

( f'(1) = 2(1 - 2)(1 + 1) + (1 - 2)^2 \cdot 1 )

( f'(1) = 2(-1)(2) + (-1)^2 \cdot 1 )

( f'(1) = -4 + 1 )

( f'(1) = -3 )

Since ( f'(1) = -3 ) is negative, ( f(x) ) is decreasing 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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