Final answer:
To find the extreme values of the function z(x,y)=xy with the condition x²+y²=1, we can use the method of Lagrange multipliers. The extreme values of the function are z(π/4) = 1/2 and z(5π/4) = -1/2.
Step-by-step explanation:
To find the extreme values of the function z(x,y) = xy with the condition x²+y²=1, we can use the method of Lagrange multipliers. Since the constraint equation is a circle, we can use the parametric equations for the circle, x = cos(t) and y = sin(t), where t is an angle between 0 and 2π.
Substituting the parametric equations into the function, we get z(t) = cos(t)sin(t). To find the extreme values, we take the derivative of z(t) with respect to t and set it equal to zero :
z'(t) = sin²(t) - cos²(t) = 0
Solving this equation, we find that there are two critical points at t = π/4 and t = 5π/4. Substituting these values back into the parametric equations, we get the extreme values of the function: z(π/4) = 1/2 and z(5π/4) = -1/2.