Question

Find the maximum value of f(x,y) = x⁴y⁸ for x, y ≥ 0. from the equation of the unit circle x²+y²=1.

Answer 1

To find the maximum value of f(x,y) = x⁴y⁸ for x, y ≥ 0 on the unit circle x²+y²=1, we can use the method of Lagrange multipliers.

Step-by-step explanation:

To find the maximum value of f(x,y) = x⁴y⁸ for x, y ≥ 0 on the unit circle x²+y²=1, we can use the method of Lagrange multipliers.

Let g(x, y) = x² + y² - 1 be the constraint function. The critical points occur when the gradient vectors of f(x, y) and g(x, y) are parallel:

∇f(x, y) = λ∇g(x, y)

Solving this system of equations will give us the maximum value of f(x, y) subject to the constraint x²+y²=1.