Is #f(x)=(x3)(x2)x^2# concave or convex at #x=3#?
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To determine if ( f(x) = (x  3)(x  2)  x^2 ) is concave or convex at ( x = 3 ), we need to examine the second derivative of the function at that point.

Find the first derivative ( f'(x) ) using the product rule. [ f'(x) = (x  3) \frac{d}{dx}(x  2) + (x  2) \frac{d}{dx}(x  3)  2x ] [ f'(x) = (x  3) + (x  2)  2x ] [ f'(x) = 2x  5 ]

Find the second derivative ( f''(x) ). [ f''(x) = \frac{d}{dx}(2x  5) ] [ f''(x) = 2 ]
Since the second derivative is constant (positive), the function is convex at ( x = 3 ).
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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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