countzander said:Homework Statement
Find the global max/min for z=xy^2 - 5 on the region bounded by y=x and y=1-x^2 in the xy-plane.
Homework Equations
The Attempt at a Solution
I found the critical point of z=xy^2 - 5 at (0,0), but I do not know how to relate this to the boundary.
countzander said:Homework Statement
Find the global max/min for z=xy^2 - 5 on the region bounded by y=x and y=1-x^2 in the xy-plane.
Homework Equations
The Attempt at a Solution
I found the critical point of z=xy^2 - 5 at (0,0), but I do not know how to relate this to the boundary.
Dick said:How did you get just (0,0) for a critical point? I think you have a whole line of critical points. Can you show us? And to handle the boundary find the intersection between the two boundary curves. Then substitute the boundary condition into your expression for z and extremize that. That will give you special points to look at along the boundary.
countzander said:dz/dx= y^2
dz/dy= 2xy
Then I set each one equal to zero and solved the system. (0,0) was the solution and so the critical point. (Critical points exist where the gradient is equal to zero.)
Will the intersection of the boundary be something like x=((sqrt5)-1)/2 or -((sqrt5)+1)/2?
Dick said:Yes, I think you've got the boundary correct. But (x,0) is also a critical point for any value of x, isn't it?
countzander said:Yes.
After plugging x into the expression for z, would I just set y=0 and solve for z?
Optimization is the process of finding the best solution to a problem, given a set of constraints and objectives. In mathematical terms, it involves finding the maximum or minimum value of a function.
The function z=xy^2 - 5 is a mathematical representation of a system or process that has two independent variables, x and y, and one dependent variable, z. It is often used in optimization problems to represent a quantity that needs to be maximized or minimized.
To find the optimal solution for z=xy^2 - 5, you can use various optimization techniques such as the gradient descent method, the Newton's method, or the simplex method. These methods involve finding the critical points of the function and evaluating them to determine the optimal solution.
Constraints in an optimization problem are restrictions or limitations that must be considered when finding the optimal solution. These can include physical, financial, or other limitations that the solution must adhere to. In the function z=xy^2 - 5, the constraints could be limits on the values of x and y, such as x>0 and y<10.
Optimization is used in a wide range of real-world applications, from engineering and physics to economics and business. It is used to design efficient systems, maximize profits, minimize costs, and improve performance. Some examples include optimizing traffic flow, designing efficient supply chains, and finding the best investment portfolio.