Question

Investigate the maxima or minima of x^3 + y^3 - 3 ax y​

Answer 1

Answer:

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Answer 2

Step-by-step explanation:

What are the maxima and minima of x3+y3−3axy?

Let f(x,y)=x3+y3−3axy.

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

⇒fx=3x2−3ay=0 and fy=3y2−3ax=0.

⇒y=x2a⇒3(x2a)2−3ax=0.

⇒x4a2−ax=0⇒x4−a3x=0.

⇒x(x3−a3)=0.

⇒x=0 or x=a.

When x=0,y=0 and when x=a,y=a.

fxx=6x,fyy=6y and fxy=−3a.

At (x,y)=(0,0),

fxxfyy−f2xy=36xy−9a2=−9a2<0.

⇒(x,y)=(0,0) is a saddle point.

At (x,y)=(a,a),

fxxfyy−f2xy=36xy−9a2=27a2>0

Further, fxx=6a>0 if a>0 and fxx=6a<0 if a<0.

⇒(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:

If a>0, the function does not have a maximum but has a local minimum at (a,a) having a value −a3.

If a<0, the function does not have a minimum but has a local maximum at (a,a) having a value −a3.

If a=0, the function does not have either a maximum or a minimum.