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: In other words, a function is convex If And Only If its Epigraph (the set of points lying on or above the Graph ) is a Convex Set . A function is also said to be strictly convex if : for any ''t'' in (0,1). PROPERTIES OF CONVEX FUNCTIONS A convex function ''f'' defined on some convex Open Interval ''C'' is Continuous on ''C'' and Differentiable at all but at most Countably Many points. If ''C'' is closed, then ''f'' may fail to be continuous at the endpoints of ''C''. A continuous function on an interval ''C'' is convex if and only if : for all ''x'' and ''y'' in ''C''. A Differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval. A Continuously Differentiable function of one variable is convex on an interval if and only if the function lies above all of its Tangents : ''f''(''y'') ≥ ''f''(''x'') + ''f'''(''x'') (''y'' − ''x'') for all ''x'' and ''y'' in the interval. A twice differentiable function of one variable is convex on an interval if and only if its second derivative is non-negative there; this gives a practical test for convexity. If its second derivative is positive then it is strictly convex, but the opposite is not true, as shown by ''f''(''x'') = ''x''4. More generally, a continuous, twice differentiable function of several variables is convex on a convex set if and only if its Hessian Matrix is Positive Semidefinite on the interior of the convex set. If two functions ''f'' and ''g'' are convex, then so is any weighted combination ''a'' ''f'' + ''b'' ''g'' with non-negative coefficients ''a'' and ''b''. Likewise, if ''f'' and ''g'' are convex, then the function max{''f'',''g''} is convex. Any Local Minimum of a convex function is also a Global Minimum . A ''strictly'' convex function will have at most one global minimum. | ||
| The | "http://wwwinformationdelightinfo/encyclopedia/entry/absolute_value" class="copylinks">Absolute Value function ''x'' is convex, even though it does not have a derivative at ''x'' = 0 |
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