## Quick Reference

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.

## Related content in Oxford Index

##### Reference entries

Users without a subscription are not able to see the full content. Please, subscribe or login to access all content.