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Global maximum of given set and extreme values

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Please help with the following mathematics problems.

(a) Let f be a differentiable functions defined on an open set U. Suppose that P is a point in U that f(P) is a maximum, i.e.
f(P) >= f(X) for all X E U
Show that grad f(P) =0
(b) Find the global maximum of the function
f(x,y)=x^3 +xy
defined on the set
S={(x,y)|-1<=x<=1, -1<=y<=1}
(c) Use the method of Language multiplier to find the extreme values of the function f given by
f(x,y,z)=(x+y+z)^2
subject to the condition
x^2+2y^2+3z^2=1

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https://brainmass.com/math/geometry-and-topology/global-maximum-given-set-extreme-values-605037

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(a) We need to define a new function g(y)=f(P+yT) where T is any number but is not equal to 0.
Since f(P) is a maximum, f(P)≥g(y) for all y and g(0)=f(P+0*T)=f(P).
Also, because f(P) is a maximum, f'(P)=0.
Since g'(y)=grad f(P)*T, g'(0)=f'(P)=0=grad f(P)*T.
Since T ia a non-zero number, grad f(P)=0

(b) We first look at the crtical points of ...

Solution Summary

The solution gives detailed steps on finding the global maximum on the given set and extreme values using Lagrange multipliers. Step by step calculations are provided.

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