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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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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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(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 ...

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