Need help with Lagrange multipliers

[Tiome_nguyen]

Hello everyone, i have 2 problems in my multivariable calculus homework that i can't solve . Please help me out, thank you so much

1/f(x,y)= [(x^2) -2y]^(0.5)
a) Find directional derivatives of f at (2,-6) in the direction of <-4,3>
b) Find equation of the tangent plane to the function f(x,y) in problem 1 at the point (2,-6)
i got the part a) and but i have no idea how to do part b)

2/
a)Use Lagrange multipliers to find max and min of function P(x,y) = x(y^2) +(x^3)y-5xy subject to the constraint boundaries xy = 4 , y = x+3 , y = x-3
b) find absolute maximum and minimum on the region bounded by xy <= 4, x-3 <= y <= x+3 .
i spent an hour for this problem but couldn't solve for it

 

[chiro]

Hello everyone, i have 2 problems in my multivariable calculus homework that i can't solve . Please help me out, thank you so much

1/f(x,y)= [(x^2) -2y]^(0.5)
a) Find directional derivatives of f at (2,-6) in the direction of <-4,3>
b) Find equation of the tangent plane to the function f(x,y) in problem 1 at the point (2,-6)
i got the part a) and but i have no idea how to do part b)

2/
a)Use Lagrange multipliers to find max and min of function P(x,y) = x(y^2) +(x^3)y-5xy subject to the constraint boundaries xy = 4 , y = x+3 , y = x-3
b) find absolute maximum and minimum on the region bounded by xy <= 4, x-3 <= y <= x+3 .
i spent an hour for this problem but couldn't solve for it

Hello Timoe_nguyen and welcome to the forums.

When we give out help here on PF we ask the posters to show any work that they have done before giving them advice. The reason we do this is because if people were just given answers then no-one benefits since the student will not understand for themselves, and also because we need to see the thought process of the individual so that we can correct any misconceptions they may have.

For question 1, what is the definition of the directional derivative and given a surface in 3-space, how do you define a plane with two linearly independent vectors that lie on the plane?

For question 2, again what is the definition of a Lagrange Multiplier, and in terms of minimums and maximums, what does this mean when we use inequalities?