We are asked to find the absolute maximum and absolute minimum of the function:

$f(x, y) = x + 5y$

subject to the constraint:

$x^2 + y^2 \leq 4$

This is the equation of a circle with radius 2, centered at the origin.

Step 1: Analyze the function inside the domain

The function $f(x, y) = x + 5y$ is linear, meaning that the maximum and minimum will likely occur on the boundary of the region defined by $x^2 + y^2 = 4$, which represents the boundary of the circle.

Step 2: Boundary case (using parameterization)

We can parameterize the boundary of the circle using polar coordinates:

• $x = 2\cos(\theta)$
• $y = 2\sin(\theta)$

Substitute these into the function $f(x, y)$:

$f(x, y) = 2\cos(\theta) + 5(2\sin(\theta)) = 2\cos(\theta) + 10\sin(\theta)$

Now, we want to find the values of $\theta$ that maximize and minimize this expression.

Step 3: Maximize and minimize the function on the boundary

To find the maximum and minimum, differentiate $f(\theta) = 2\cos(\theta) + 10\sin(\theta)$ with respect to $\theta$:

$\frac{d}{d\theta} \left( 2\cos(\theta) + 10\sin(\theta) \right) = -2\sin(\theta) + 10\cos(\theta)$

Set the derivative equal to zero to find critical points:

$-2\sin(\theta) + 10\cos(\theta) = 0$

Solve for $\theta$:

$\sin(\theta) = \frac{5}{\cos(\theta)}$

This simplifies to $\tan(\theta) = 5$, so:

$\theta = \tan^{-1}(5)$

Step 4: Evaluate the function at critical points and boundaries