Final answer:
To find the relative maximum value of f(x, y) subject to a constraint, identify the constraint equation and express f in terms of a single variable. Find the critical points of f(x) and evaluate f(x) at each critical point and the endpoints of the constraint interval.
Step-by-step explanation:
Relative Maximum Value of f(x, y) Subject to Constraint
To find the relative maximum value of f(x, y) subject to a constraint, follow these steps:
- Identify the constraint equation.
- Use the constraint equation to eliminate one variable in the function f(x, y) and express f in terms of a single variable.
- Find the critical points of f(x) by taking the derivative and setting it equal to zero.
- Evaluate f(x) at each critical point to determine the maximum value.
Remember to check for endpoints of the constraint interval as well.
For example, if the constraint equation is x + y = 10 and the function is f(x, y) = x^2 + y^2, you can solve the constraint equation for y in terms of x to get y = 10 - x. Substitute this expression into f(x, y) to get f(x) = x^2 + (10 - x)^2. Take the derivative, set it equal to zero, and find the x values of the critical points. Evaluate f(x) at these critical points and the endpoints of the constraint interval to determine the relative maximum value.