# What are the points of inflection of #f(x)=2xe^-x + x^2e^x #?

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To find the points of inflection for the function ( f(x) = 2xe^{-x} + x^2e^x ), we first need to find its second derivative, ( f''(x) ), and then solve for ( x ) where ( f''(x) = 0 ).

First derivative: [ f'(x) = 2e^{-x} - 2xe^{-x} + 2xe^x + 2x^2e^x ]

Second derivative: [ f''(x) = -2e^{-x} - 2xe^{-x} - 2xe^x + 2e^x + 2x^2e^x + 2x^2e^x ]

Simplify the second derivative: [ f''(x) = -2e^{-x} - 2xe^{-x} - 2xe^x + 2e^x + 4x^2e^x ]

Now, set ( f''(x) ) equal to zero and solve for ( x ): [ -2e^{-x} - 2xe^{-x} - 2xe^x + 2e^x + 4x^2e^x = 0 ]

There is no simple analytical solution to this equation. You would typically use numerical methods to approximate the solutions. Once you find the ( x )-values where the second derivative equals zero, you can test these values to determine whether they correspond to points of inflection by checking the concavity of the function around those points. If the concavity changes at those points, they are points of inflection.

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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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