Absolute minimum and maximum
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1.)f(x,y)= 2x^3 + y^4, D={(x,y), x^2 + y^2 less than or equal to 1}
2.)f(x,y)= x^3 - 3x - y^3 + 12y, D is the quadrilateral whose vertices are (-2,3), (2,3), (2,2), and (-2,-2).
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Solution Summary
This solution provides two examples of finding absolute maximum and minimum values of a function.
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** Please see the attached file for the complete solution response **
Any smooth function on a compact set has an absolute maximum and minimum. These can only occur at critical points, namely:
a) Stationary points where the partial derivatives are zero.
b) Boundary points of the domain.
Since there definitely is an absolute max and an absolute min, if we can find the location of all the critical points, then simply comparing the values of the function at these points gives the required answer.
First find the interior stationary points by setting the partial derivatives to zero and solving simultaneously. Be very careful when you do this: it is entirely possible you will find stationary points for the function which lie outside the domain of consideration. These must be discarded.
Next, try to use the boundary condition to express the function (restricted to the ...
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