Find the extreme values of the function f(x,y) = x2 + 2y2 on the circl | Mathematics I 2075

Mathematics I20755 marksNumerical: Lagrange multipliers extremaMaximum and minimum values

Find the extreme values of the function $f(x,y) = x^2 + 2y^2$ on the circle $x^2 + y^2 = 1$ . [5]

Find the extreme values of the function $f(x,y) = x^2 + 2y^2$ on the circle $x^2 + y^2 = 1$. [5]

Objective function: $f(x,y) = x^2 + 2y^2$ - Constraint: $g(x,y) = x^2 + y^2 - 1 = 0$ Set $\nabla f = \lambda \nabla g$ . $fx = 2x,\quad fy = 4y,\qquad gx = 2x,\quad gy = 2y$ System of equations: $2x = \lambda(2x) \;\Rightarrow\; 2x(1-\lambda) = 0 \tag{1}$ $$ 4y = \lambda(2y) ;\Rightarrow...

Objective function: $f(x,y) = x^2 + 2y^2$ - Constraint: $g(x,y) = x^2 + y^2 - 1 = 0$ Set $\nabla f = \lambda \nabla g$. $$fx = 2x,\quad fy = 4y,\qquad gx = 2x,\quad gy = 2y$$ System of equations: $$ 2x = \lambda(2x) ;\Rightarrow; 2x(1-\lambda) = 0 \tag{1} $$ $$ 4y = \lambda(2y) ;\Rightarrow...

Find the extreme values of the function f(x,y) = x2 + 2y2 on the circle x2 + y2 = 1.