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Given a function with $2$ variable $f(x, y)$ , I need a general rule to find max and min of this function in a range for $a\leq x\leq b, c\leq y\leq d$.

I know how to differentiate two variable function but that would only give local maximum and minimum which may lie outside the range.

TheHolyJoker
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Roushan Singh
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    Googling "how to find maxima and minima in an interval" gives you several such tutorials. Do you have a more specific question? – Simply Beautiful Art Dec 18 '19 at 14:29
  • You have to check the boundary as well, where it becomes one-variable optimization problem since you know the value of the other variable as your boundary is a square. – gt6989b Dec 18 '19 at 14:29
  • lets suppose f(x, y) = cos(x)*cos(x) + sin(y). now i need the maximum and minimum in range 30 deg <= x <= 60 deg and 10 deg <= y <= 40 deg. i would really appreciate if you can use this specific case to explain the solution – Roushan Singh Dec 18 '19 at 14:32
  • That is not a good example as it is almost trivial: $\cos^2 x$ is strictly decreasing on its interval and $\sin y$ is strictly increasing on its interval, so the max is at $x = 30^\circ, y = 40^\circ$ and the min is at $x = 60^\circ, y = 10^\circ$. – Paul Sinclair Dec 18 '19 at 22:58

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