Question

For the function f(x,y) = x2 + y2 - xy subject to the contrsiant

x - y = 10, what is the solution of x and y in point form (x,y) to
system? Also, what is the value of f(x,y) at that point?.

Answer 1

The solution to the constrained system is (5, -5) for the variables x and y respectively, and the value of the function f(x,y) at this point is 50.

Step-by-step explanation:

The student asks for the solution of x and y for the function f(x,y) = x2 + y2 - xy under the constraint x - y = 10. To find the solution, we shall express y in terms of x using the constraint equation and then substitute this expression into the function f(x,y).

From the constraint equation we have y = x - 10. Substituting this into the function gives f(x, x-10) which becomes f(x) = x2 + (x-10)2 - x(x-10). Simplifying, f(x) = 2x2 - 20x + 100. To find the minimum value, we can take the derivative of f(x) with respect to x and set it to zero since it's a parabola that opens upwards.

Derivative: f'(x) = 4x - 20 = 0.
Solving this gives x = 5. Plugging x = 5 back into the constraint yields y = -5. The solution in point form is (5, -5) and the value of f(x,y) at that point is 50.