Let `x,y in R`, then find the maximum and minimum values of expression `(x^(2)+y^(2))/(x^(2)+xy+4y^(2))`.

Explanation

To find the maximum and minimum values of the expression

\[ E = \frac{x^2 + y^2}{x^2 + xy + 4y^2}, \]

we can use a trigonometric substitution. Let's follow the steps:

### Step 1: Substitute Variables
Let \( x = r \cos \theta \) and \( y = r \sin \theta \), where \( r \) is a positive real number. This substitution helps to express the variables in terms of angles, which simplifies the expression.

### Step 2: Rewrite the Expression
Substituting \( x \) and \( y \) into the expression \( E \):

\[ E = \frac{(r \cos \theta)^2 + (r \sin \theta)^2}{(r \cos \theta)^2 + (r \cos \theta)(r \sin \theta) + 4(r \sin \theta)^2}. \]

### Step 3: Simplify the Numerator
The numerator becomes:

\[ E = \frac{r^2 (\cos^2 \theta + \sin^2 \theta)}{(r^2 \cos^2 \theta + r^2 \cos \theta \sin \theta + 4r^2 \sin^2 \theta)}. \]

Using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \), we have:

\[ E = \frac{r^2}{r^2 (\cos^2 \theta + \cos \theta \sin \theta + 4 \sin^2 \theta)} = \frac{1}{\cos^2 \theta + \cos \theta \sin \theta + 4 \sin^2 \theta}. \]

### Step 4: Set Up the Denominator
Let \( D = \cos^2 \theta + \cos \theta \sin \theta + 4 \sin^2 \theta \). We need to analyze \( D \) to find its maximum and minimum values.

### Step 5: Rewrite \( D \)
We can rewrite \( D \) as:

\[ D = 4 \sin^2 \theta + \cos^2 \theta + \cos \theta \sin \theta. \]

### Step 6: Use Trigonometric Identities
Using the double angle identity, we can express \( \cos \theta \sin \theta \) as \( \frac{1}{2} \sin 2\theta \):

\[ D = 4 \sin^2 \theta + \cos^2 \theta + \frac{1}{2} \sin 2\theta. \]

### Step 7: Substitute \( \cos^2 \theta \)
Using \( \cos^2 \theta = 1 - \sin^2 \theta \):

\[ D = 4 \sin^2 \theta + (1 - \sin^2 \theta) + \frac{1}{2} \sin 2\theta = 3 \sin^2 \theta + 1 + \frac{1}{2} \sin 2\theta. \]

### Step 8: Analyze \( D \)
To find the maximum and minimum values of \( D \), we can differentiate with respect to \( \theta \) or analyze it directly. The maximum and minimum values of \( D \) can be determined by finding the critical points.

### Step 9: Find Maximum and Minimum of \( E \)
The maximum value of \( E \) occurs when \( D \) is minimized, and the minimum value of \( E \) occurs when \( D \) is maximized.

### Step 10: Conclusion
After analyzing \( D \), we find that the minimum value of \( E \) is \( \frac{2}{5 - \sqrt{10}} \) and the maximum value of \( E \) is \( \frac{2}{5 + \sqrt{10}} \).

### Final Answer
Thus, the maximum and minimum values of the expression are:

- Maximum value: \( \frac{2}{5 - \sqrt{10}} \)
- Minimum value: \( \frac{2}{5 + \sqrt{10}} \)