# For what values of x is #f(x)=(x+2)(x-3)(x+1)# concave or convex?

The function is concave for

The function is convex for

We need

We must calculate the second derivative and determine the sign.

Therefore,

This is the point of inflexion.

We can build a chart

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To determine the concavity of ( f(x) = (x + 2)(x - 3)(x + 1) ), we need to examine its second derivative. If the second derivative is positive, the function is concave upward (convex), and if it's negative, the function is concave downward.

First, find the first derivative:

[ f'(x) = (x - 3)(x + 1) + (x + 2)(x + 1) + (x + 2)(x - 3) ]

Then, find the second derivative:

[ f''(x) = 2(x - 3) + 2(x + 1) + 2(x + 2) ]

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

[ f''(x) = 6x ]

Since the coefficient of ( x ) in the second derivative ( 6x ) is positive for all real values of ( x ), the function ( f(x) ) is concave upward (convex) for all ( x ).

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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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