Answer
See explanation
Work Step by Step
Step 1: Lagrange Multipliers Using Lagrange multipliers for a two-variable function \(f\) and one constraint \(g\), we get: \(\nabla f = \lambda \nabla g\) Geometrically, this is because at an extremum, \(\nabla f\) and \(\nabla g\) are parallel. Step 2: Equations from Lagrange Multipliers To fulfill the vector equality, we get two equations: \[ \begin{cases} f_x(x, y) = \lambda g_x(x, y) \\ f_y(x, y) = \lambda g_y(x, y) \end{cases} \] Then, we cancel \(\lambda\) to get a single equation expressing \(y\) in function of \(x\). Geometrically, this equation represents a function. Step 3: Substitution into Constraint Next, we need to substitute the expression of \(y\) in terms of \(x\) into the constraint equation: \[ g(x, y) = 0 \] This will result in one or multiple points representing the extremum. Geometrically, this represents the points of intersection between the equation of the constraint \(g(x, y) = 0\) and the equation we get from Lagrange multipliers. Step 4: Evaluate the Function Finally, we substitute the points of intersection into the function \(f\), and according to the value we get, we can draw conclusions about maximum and minimum points.
