{"id":4455,"date":"2013-12-23T06:05:22","date_gmt":"2013-12-22T21:05:22","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4455"},"modified":"2014-08-01T19:09:22","modified_gmt":"2014-08-01T10:09:22","slug":"maxima-and-minima","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4455","title":{"rendered":"Maxima and minima"},"content":{"rendered":"
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If for all x<\/span> such that |x – a<\/span>| < δ<\/span> where f(x)<\/span> ≤ f(a)<\/span> [ or f(x)<\/span> ≥ f(a)<\/span>], f(a)<\/span> is a relative maximam<\/span> [ or relative minimum<\/span>]. For f(x)<\/span> to have a relative maximum or minimum at x = a<\/span>, it must have f'(a)<\/span> = 0. Then if f”(a)<\/span> < 0 it is a relative maximum while if f”(a)<\/span> ≥ 0 it is a relative minimum. Possible points at which f(x)<\/span> has a relative maxima or minima are obtained by solving f'(x)<\/span> = 0, i.e. by finding the values of x<\/span> where the slope<\/span> of the graph f(x)<\/span> is equal to zero. <\/p>\n

Similarly f(x, y)<\/span> has a relative maximum or minimum at x = a, y = b<\/span> if fx<\/sub>(a, b)<\/span> = 0, fy<\/sub>(a, b)<\/span> = 0. Thus possible points at which f(x, y)<\/span> has relative maxima or minima are obtained by solving simultaneously the equations<\/p>\n

\\displaystyle \\frac{\\partial f}{\\partial x} = 0,\\ \\frac{\\partial f}{\\partial y} = 0<\/p>\n

Extensions to functions of more than two variables are similar. <\/p>\n<\/blockquote>\n