Question

Find absolute maximum and minimum values for f (x, y) = x2 + 10xy + y^2 defined on the disc D = x² + y2 <3.

Answer 1

The maximum value is 81 at (3√2, 3√2) and the minimum value is 0 at the origin (0, 0). To find the absolute maximum and minimum values of the given function over the disc D, we check the critical points in the interior and the points on the boundary.

Step-by-step explanation:

In order to identify the absolute maximum and minimum values of a function over a closed region, a thorough examination of critical points within the region and points on the boundary is necessary.

Initially, critical points are pinpointed by setting the partial derivatives of f(x, y) to zero and solving for x and y, leading to the coordinates x = -5 and y = 5.

Subsequently, scrutiny extends to points on the boundary of the region, which, in this case, is defined as the disc x² + y² < 3.

By substituting the boundary equation x² + y² = 3 into the function f(x, y), computations reveal the maximum value of 81 at the point (3√2, 3√2) and the minimum value of 0 at the origin (0, 0).

This comprehensive analysis ensures a comprehensive understanding of the function's behavior over both interior and boundary points.

Answer 2

The absolute maximum value is
`\( (3)/(2) \)`.

Let's solve the system of equations to find the critical points using the method of Lagrange multipliers.

1. Set up the Lagrangian function:


`\[ L(x, y, \lambda) = x^2 + 10xy + y^2 + \lambda(3 - x^2 - y^2) \]`

2. Take the partial derivatives and set them equal to zero:


`\[ (\partial L)/(\partial x) = 2x + 10y - 2\lambda x = 0 \]\[ (\partial L)/(\partial y) = 10x + 2y - 2\lambda y = 0 \]\[ (\partial L)/(\partial \lambda) = 3 - x^2 - y^2 = 0 \]`

Solving these equations simultaneously, we get the critical points:


`\[ x = (1)/(√(2)), \quad y = (1)/(√(2)), \quad \lambda = (1)/(2) \]`

Now, evaluate \( f(x, y) \) at the critical point:


`\[ f\left((1)/(√(2)), (1)/(√(2))\right) = (3)/(2) \]\\`
Next, consider the boundary of
`\( D \),` where \( x^2 + y^2 = 3 \). Parametrize the boundary using polar coordinates:


`\[ x(t) = √(3)\cos(t), \quad y(t) = √(3)\sin(t), \quad 0 \leq t \leq 2\pi \]`

Evaluate
`\( f(x, y) \)`on the boundary:

`\[ f(x(t), y(t)) = 3 + 30√(3)\sin(t)\cos(t) \]`

Find the minimum and maximum values of this expression by considering critical points and endpoints of the parameterization.

The absolute maximum value is
`\( (3)/(2) \)`at the critical point, and the absolute minimum and maximum values on the boundary are obtained by evaluating
`\( f(x, y) \)`at the corresponding points in the parametrization.