Find the absolute maximum and minimum of f(x,y)=x+5y 
 within the domain x^2+y^2<=4 

Absolute minimum of 
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Absolute maximum of 
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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



Find the absolute maximum and minimum of f(x,y)=x+5y within the domain x^2 + y^2 <= 4.

This problem involves finding the absolute maximum and minimum values of the function f(x, y) = x + 5y within a circular domain defined by x² + y² ≤ 4. The solution includes using polar coordinates, analyzing boundary conditions, and applying derivative techniques for optimization. Ideal for undergraduate students studying multivariable calculus.

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