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concavity


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At a point of a graph y=f(x), it may be possible to specify the concavity by describing the curve as either concave up or concave down at that point, as follows,If the second derivative f″(x) exists and is positive throughout some neighbourhood of a point a, then f′(x) is strictly increasing in that neighbourhood, and the curve is said to be concave up at a. At that point, the graph y=f(x) and its tangent look like one of the cases shown in the first figure. If f″(a)>0 and f″ is continuous at a, it follows that y=f(x) is concave up at a. Consequently, if f′(a)=0 and f″(a)>0, the function f has a local minimum at a. Similarly, if f″(x) exists and is negative throughout some neighbourhood of a, or if f″(a)<0 and f″ is continuous at a, then the graph y=f(x) is concave down at a and looks like one of the cases shown in the second figure. If f′(a)=0 and f″(a)<0, the function f has a local maximum at a.

Subjects: Mathematics.


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