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

Answer 1

Isaiah Allen

#f# is a linear function. It is neither concave nor convex.

Simplify to see that #f(x) = -5x+6#.

If more is needed, observe that #f'(x) = -5# and #f''(x) = 0#, so #f# is neither concave nor convex.

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Answer 2

Cameron Alexander

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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Answer from HIX Tutor

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.