# Is #f(x)=x^3-xe^(x-x^2) # concave or convex at #x=1 #?

To find intervals of convex and concave, we have to find the 2nd derivative.

graph{6x-(-2x^2+x)(-2x+1)(e^(-x^2+x))-(-4x+1)(e^(-x^2+x))-(-2x+1)(e^(-x^2+x)) [-10, 10, -5, 5]}

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To determine if ( f(x) = x^3 - xe^{x - x^2} ) is concave or convex at ( x = 1 ), we need to examine the second derivative test. If the second derivative is positive, the function is concave up (convex); if it's negative, the function is concave down.

First, find the second derivative of ( f(x) ):

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

[ f''(x) = 6x - e^{x - x^2} + 2xe^{x - x^2} - 4x^2e^{x - x^2} ]

Now, evaluate ( f''(1) ):

[ f''(1) = 6(1) - e^{1 - 1} + 2(1)e^{1 - 1} - 4(1)^2e^{1 - 1} ]

[ f''(1) = 6 - e^0 + 2e^0 - 4e^0 ]

[ f''(1) = 6 - 1 + 2 - 4 ]

[ f''(1) = 3 ]

Since the second derivative ( f''(1) ) is positive, the function ( f(x) = x^3 - xe^{x - x^2} ) is concave up (convex) 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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