Lagrange Multiplier theory question

[flyingpig]

Homework Statement

I made this up, so I am not even sure if there is a solution

Let's say I have to find values for which these two inequality hold $$x^2 + y^5 + z = 6$$ and $$8xy + z^9 \sin(x) + 2yx \leq 200$$And by Lagrange Multipliers that

$$\nabla f = \mu \nabla g$$

So can I let $$f = 8xy + z^9 \sin(x) + 2yx - 200 \leq 0$$?

 

[HallsofIvy]

What question are you trying to answer? Normally, the Lagrange multiplier method is used to find a maximum or minimum to a given function with some additional constraints. Here, you don't seem to have any function to maximize or minimize

 

[flyingpig]

What question are you trying to answer? Normally, the Lagrange multiplier method is used to find a maximum or minimum to a given function with some additional constraints. Here, you don't seem to have any function to maximize or minimize

Yeah I made one by making f <= 0...?

Okay I got this idea from another problem from this video



go to 5:05...

 

[Ray Vickson]

I guess you want to know if it is possible to find some x,y,x that give you f(x,y,z) <= 0 and g(x,y,x) = 0, where f and g are the two functions given in your post. One way would be to solve the problem
minimize f, subject to g = 0, then check if the min value of f is <= 0. This approach is pretty standard, for example, when checking if a set of linear equations and inequalities is feasible.

RGV