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maximum value of the function f(x,y)=x3+y3+3xy is

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Let  f(x,y)=x3+y3−3axy.

At the critical values of the function, both the first derivatives are zero.

⇒ (x,y) = (a,a)  represents a minimum if  a>0  and represents a maximum if  a<0.

When  (x,y)=(a,a),f(x,y)=x3+y3−3axy=−a3.

If  a=0,  the function becomes  x3+y3, which does not have any maximum or minimum and has a saddle point at  (x,y)=(0,0).

Therefore, we conclude as under:

1. If a>0,  the function does not have a maximum but has a local minimum at  (a,a) having a value  −a3.
2. If a<0, the function does not have a minimum but has a local maximum at  (a,a) having a value  −a3.
3. If  a=0, the function does not have either a maximum or a minimum.

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