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Calculus 3 Maximizations Problem (Need major help with this question!)

I have:

f(x,y): http://www.wolframalpha.com/input/?i=(x^6+*+y^4+*+(4+-+x+-+y)^7)

∂f/∂x: http://www.wolframalpha.com/input/?i=-x^5+(13+x%2B6+(-4%2By))+y^4+(-4%2Bx%2By)^6+%3D+0,+Solve+for+x

∂f/∂y: http://www.wolframalpha.com/input/?i=-x^6+y^3+(-4%2Bx%2By)^6+(-16%2B4+x%2B11+y)+%3D+0,+Solve+for+x

∂^2 f/(∂x ∂y): http://www.wolframalpha.com/input/?i=-2+x^5+y^3+(-4%2Bx%2By)^5+(192-152+x%2B26+x^2-180+y%2B80+x+y%2B33+y^2)+%3D+0,+Solve+for+x

where I told Wolfram Alpha to isolate x for each equation with partial derivatives because I think that's what I need to do in order to get the critical points.

I'm confused as to what I need to do next. I heard about a Hessian matrix method but I don't know if that's necessary or if I can just do a regular system of equations. I read the Paul's Online Notes website but I'm still stuck for this particular problem.

What do I do when I solve for x when each of the equations are equal to 0? If anything is unclear, just ask me to clarify it.

Any input would be appreciated!

Thanks in advance!

Re: Calculus 3 Maximizations Problem (Need major help with this question!)

Quote:

Originally Posted by

**s3a** I have:

f(x,y): http://www.wolframalpha.com/input/?i=(x^6+*+y^4+*+(4+-+x+-+y)^7)

∂f/∂x: http://www.wolframalpha.com/input/?i=-x^5+(13+x%2B6+(-4%2By))+y^4+(-4%2Bx%2By)^6+%3D+0,+Solve+for+x

∂f/∂y: http://www.wolframalpha.com/input/?i=-x^6+y^3+(-4%2Bx%2By)^6+(-16%2B4+x%2B11+y)+%3D+0,+Solve+for+x

∂^2 f/(∂x ∂y): http://www.wolframalpha.com/input/?i=-2+x^5+y^3+(-4%2Bx%2By)^5+(192-152+x%2B26+x^2-180+y%2B80+x+y%2B33+y^2)+%3D+0,+Solve+for+x

where I told Wolfram Alpha to isolate x for each equation with partial derivatives because I think that's what I need to do in order to get the critical points.

I'm confused as to what I need to do next. I heard about a Hessian matrix method but I don't know if that's necessary or if I can just do a regular system of equations. I read the Paul's Online Notes website but I'm still stuck for this particular problem.

What do I do when I solve for x when each of the equations are equal to 0? If anything is unclear, just ask me to clarify it.

Any input would be appreciated!

Thanks in advance!

The two partial derivatives when set to zero can be reduced to a pair of linear equations in x and y. Have you solved them?

That point (if it is in the feasible region) will be either a maximum, minimum or a saddle which you will need to classify. If it is not in the feasible region the maximum will occur on the boundary of the feasible region.

CB

Re: Calculus 3 Maximizations Problem (Need major help with this question!)

I ended up getting help for it after extensive search before this but I do like the way you summarized it so it works as a nice refresher for me so thank you :).