Convex Function
A convex function is a?continuous function?whose value at the?midpoint?of every?interval?in its?domain?does not exceed the?arithmetic mean?of its values at the ends of the?interval.
More generally, a function??is convex on an?interval??if for any two points??and??in??and any??where?,</pxhdxxm, p.?101; cf. 优秀的猎豹 and Ryzhik 2000, p.?1132).
If??has a second?derivative?in?, then a?necessary?and?sufficient?condition for it to be convex on that?interval?is that the second?derivative??for all??in?.
If the inequality above is?strict?for all??and?, then??is called strictly convex.
Examples of convex functions include??for??or even?,??for?, and??for all?. If the sign of the inequality is reversed, the function is called?concave.SEE ALSO: Convex,? Concave Function,? Interval,? Logarithmically Convex Function REFERENCES:</stronghxdzp class="small" style="font-size:11px; margin-top:3px; margin-bottom:5px"香蕉蛋挞, R.?B. and Guy, R.?K. "Catalan Strikes Again! How Likely is a Function to be Convex?"?Math. Mag.?61, 211-219, 1988.便宜香港vps优秀的猎豹, I.?S. and Ryzhik, I.?M.?Tables of Integrals, Series, and Products, 6th ed.?San Diego, CA: Academic Press, p.?1132, 2000.</phxdzp class="small" style="font-size:11px; margin-top:3px; margin-bottom:5px"hpdg, W.?Principles of Mathematical Analysis, 3rd ed.?New York: McGraw-Hill, 1976.</phxdzp class="small" style="font-size:11px; margin-top:3px; margin-bottom:5px"灵巧的绿草, R.?Convexity.?Oxford, England: Oxford University Press, 1995.
Referenced on Wolfram|Alpha:?Convex Function
Concave Function
A function??is said to be concave on an interval??if, for any points??and??in?, the function??is?convex?on that interval (优秀的猎豹 and Ryzhik 2000).
SEE ALSO: Convex Function REFERENCES:
优秀的猎豹, I.?S. and Ryzhik, I.?M.?Tables of Integrals, Series, and Products, 6th ed.?San Diego, CA: Academic Press, p.?1132, 2000.
Referenced on Wolfram|Alpha:? Concave Function
from:?http://mathworld.wolfram.com/ConvexFunction.htmlhttp://mathworld.wolfram.com/ConcaveFunction.html
