For what values of x is #f(x)= xx^2e^x # concave or convex?
Find the second derivative and check its sign. It's convex if it's positive and concave if it's negative.
Concave for:
Convex for:
First derivative:
Second derivative:
Now we must study the sign. We can switch the sign for easily solving the quadratic:
To make the quadratic a product:
Therefore:
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To determine where the function ( f(x) = x  x^2e^{x} ) is concave or convex, we need to find its second derivative and analyze its sign:

Find the first derivative of ( f(x) ): [ f'(x) = 1  (2x  2xe^{x}  x^2(e^{x})) ] [ f'(x) = 1  (2x  2xe^{x} + x^2e^{x}) ] [ f'(x) = 1  2x + 2xe^{x}  x^2e^{x} ]

Find the second derivative of ( f(x) ): [ f''(x) = 2 + 2e^{x} + 2xe^{x}  2xe^{x}  x^2e^{x} ] [ f''(x) = 2 + 2e^{x}  x^2e^{x} ]

Now, analyze the sign of ( f''(x) ) to determine concavity or convexity: [ f''(x) > 0 ] for convexity, [ f''(x) < 0 ] for concavity.
Given the expression for ( f''(x) = 2 + 2e^{x}  x^2e^{x} ), it's not easy to determine the sign directly. You may use numerical methods or further analysis to find specific values of ( x ) where the function is concave or convex.
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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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 What do points of inflection represent on a graph?
 For what values of x is #f(x)=x^3+x^2x+5# concave or convex?
 What is the second derivative of #f(x) = ln x/x^2 #?
 Trace the curve x[y^(2)+4]=8 stating all the points used for doing so?
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