Mathematics

Mathematics

Problem

For f(x)=x3−3x2+2f(x)=x^{3}-3x^{2}+2, determine the concavity of the graph at x=1x=1. a) Concave up at x=1x=1 because f′′(1)>0f''(1)>0. b) Neither concave up nor concave down at x=1;  f′′(1)=0x=1;\;f''(1)=0 and f′′f'' changes sign. c) Concave down at x=1x=1 because f′(1)<0f'(1)<0. d) Cannot be determined because f′′(1)=0f''(1)=0 but f′′f'' does not change sign.
Intuitive explanation

Concavity is determined by the second derivative—it tells you whether the curve is bending upward or downward. Find f′′(x)f''(x), evaluate it at x=1x=1, and check the sign: positive means concave up (like a cup), negative means concave down (like an arch). This single number reveals the shape of the graph at that point.

Intuitive explanation

Concavity is determined by the second derivative—it tells you whether the curve is bending upward or downward. Find f′′(x)f''(x), evaluate it at x=1x=1, and check the sign: positive means concave up (like a cup), negative means concave down (like an arch). This single number reveals the shape of the graph at that point.

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