Often in calculus you are taught how to locate maxima and minima of functions of two variables using contour plots. I would like to write down and prove a precise statement about this way of identifying critical points. What kind of hypotheses do we need on the function? What is the exact meaning of sentences such as "all contours increase/decrease as we move toward the maximum/minimum"?
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A contour is usually a curve on the plane and I have no idea what 'all contours (i.e. those curves) increase' might mean. I suppose it means 'whatever direction we choose, when approaching the maximum, we meet contours of increasing values' — but that's false, imho. The function $f(t)$, where $t$ is a position at a chosen line, needn't be strictly increasing on any interval $(t_0,t_{max}]$ to reach a maximum at $t_{max}$. Anyway, we needn't approach maximum along a straight line (move towards it) – we can approach it e.g. by a spiral... – CiaPan Jan 03 '18 at 14:00
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I've been using a home-made contouring program as a matter of routine while answering questions here at MSE. See e.g. Solving for streamlines from numerical velocity field . And the links to my website in there : MSE publications / references 2016 , especially contours.pas and PaulBourke.pas . – Han de Bruijn Jan 03 '18 at 19:26
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Five (greatly enlarged) screen shots illustrating the idea starting here – Han de Bruijn Jan 03 '18 at 19:36
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Applied to finding extreme values, two examples. First : $a^{|b-a|}+b^{|c-b|}+c^{|a-c|} > \frac52$ for $a,b,c >0$ and $a+b+c=3$ . Second : Prove $\frac{xy}{5y^3+4}+\frac{yz}{5z^3+4}+\frac{zx}{5x^3+4} \leqslant \frac13$ – Han de Bruijn Jan 03 '18 at 20:15
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Please, if you are ok, you can accept the answer and set it as solved. Thanks! – user Feb 02 '18 at 21:36
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It is a convenient method to find maxima and minima only for functions
$$f(x,y): D\subseteq \mathbb{R^2}\to \mathbb{R}$$
and only when the contur plots $f(x,y)=k$ have a simple representation in $\mathbb{R^2}$.
In this case the method consists simply to drawn in the plane the domain $D$ and the contour plots and by inspection to find the points of contact at wich $k_{max}$ and $k_{min}$ are attained.
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