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Finding global extrema and checking that we have actually found one is harder than finding local extrema, in general. There is one nice case: that of convex and concave functions. A convex function is one where the line segment connecting two points (x,f(x)) and (y,f(y)) lies above the function. Mathematically, a function f is convex if, for all x, y and all tex2html_wrap_inline938 ,


See Figure 1.3. The function f is concave if -f is convex.

Figure 1.3: Convex function and concave function

There is an easy way to check for convexity when f is twice differentiable: the function f is convex on some domain [a,b] if (and only if) tex2html_wrap_inline952 for all x in the domain. Similarly, f is concave on some domain [a,b] if (and only if) tex2html_wrap_inline960 for all x in the domain.

If f(x) is convex, then any local minimum is also a global minimum.

If f(x) is concave, then any local maximum is also a global maximum.





Michael A. Trick
Mon Aug 24 13:43:30 EDT 1998