Is #f(x)=4xe^(3x2) # increasing or decreasing at #x=2 #?
Differentiating the function (this is where the chain rule comes in):
graph{[13.19, 12.12, 6.53, 6.13]}e^(3x2)
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To determine whether ( f(x) = 4x  e^{3x  2} ) is increasing or decreasing at ( x = 2 ), you need to evaluate the sign of the derivative of ( f(x) ) at that point. If the derivative is positive, the function is increasing; if it's negative, the function is decreasing.

Find the derivative of ( f(x) ) with respect to ( x ): [ f'(x) = 4  3e^{3x  2} ]

Evaluate the derivative at ( x = 2 ): [ f'(2) = 4  3e^{3(2)  2} = 4  3e^{8} ]

Determine the sign of ( f'(2) ): Since ( e^{8} ) is positive (since it's the reciprocal of a large positive number), ( f'(2) ) will be positive.
Conclusion: At ( x = 2 ), ( f(x) ) is increasing.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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