How do you find the critical points and the open intervals where the function is increasing and decreasing for #y = xe^(x(2  3x))#?
The function is increasing on the interval:
Jasivan S, gives an excellent answer to the question asked, but I suspect that the intended question was: Find the open intervals on which the function is increasing and those on which it is decreasing. (Rather than finding those on which it is doing both.)
Cosequently,
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To find the critical points and the intervals where the function is increasing or decreasing for ( y = xe^{x(2  3x)} ), follow these steps:
 Find the derivative of the function ( y ) with respect to ( x ), denoted as ( y' ).
 Set ( y' ) equal to zero and solve for ( x ) to find the critical points.
 Determine the intervals where ( y' > 0 ) to identify where the function is increasing.
 Determine the intervals where ( y' < 0 ) to identify where the function is decreasing.
Let's go through these steps:
 The derivative of the function ( y = xe^{x(2  3x)} ) can be found using the product rule and chain rule:
[ y' = e^{x(2  3x)} + x(e^{x(2  3x)})\cdot(2  3x) ]
 Set ( y' ) equal to zero and solve for ( x ) to find the critical points:
[ e^{x(2  3x)} + x(e^{x(2  3x)})\cdot(2  3x) = 0 ]
This equation may not have an explicit solution, so you may need to use numerical methods or graphing techniques to approximate the critical points.

Determine the intervals where ( y' > 0 ) to identify where the function is increasing.

Determine the intervals where ( y' < 0 ) to identify where the function is decreasing.
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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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