# Is #f(x)=-e^(x^2-3x+2) # increasing or decreasing at #x=0 #?

will be an increasing function

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To determine if ( f(x) = -e^{x^2 - 3x + 2} ) is increasing or decreasing at ( x = 0 ), we need to examine the sign of the derivative ( f'(x) ) at ( x = 0 ).

Calculate the derivative ( f'(x) ) using the chain rule:

[ f'(x) = -e^{x^2 - 3x + 2} \cdot \frac{d}{dx}(x^2 - 3x + 2) ]

[ f'(x) = -e^{x^2 - 3x + 2} \cdot (2x - 3) ]

Now, evaluate ( f'(0) ):

[ f'(0) = -e^{0^2 - 3(0) + 2} \cdot (2(0) - 3) ]

[ f'(0) = -e^2 \cdot (-3) ]

Since ( e^2 ) is a positive constant, and multiplying it by (-3) gives a negative result, ( f'(0) ) is negative.

Therefore, since the derivative ( f'(x) ) is negative at ( x = 0 ), the function ( f(x) = -e^{x^2 - 3x + 2} ) 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.

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