given x,y are real numbers,x^2+y^2>0. if the maximum and minimum value of the expression E=x^2+y^2/x^2+xy+4y^2 are M and m, and A denotes the average value of M and m, compute A
To find the maximum and minimum values of the expression E = x^2 + y^2 / (x^2 + xy + 4y^2), we need to analyze the function and determine its critical points.
First, let's find the critical points by taking the partial derivatives with respect to x and y and setting them equal to zero.
∂E/∂x = (2x(x^2 + xy + 4y^2) - 2(x^2 + y^2)(2x + y)) / (x^2 + xy + 4y^2)^2 = 0
∂E/∂y = (2y(x^2 + xy + 4y^2) - 2(x^2 + y^2)y) / (x^2 + xy + 4y^2)^2 =0
Simplifying the above equations:
2x(x^2 + xy + 4y^2) - 2(x^2 + y^2)(2x + y) = 0 [1]
2y(x^2 + xy + 4y^2) - 2(x^2 + y^2)y = 0 [2]
To solve these equations, we can start by factoring out common terms:
2(x^2 + xy + 4y^2)(x - y) = 0
(x^2 + xy + 4y^2)(x - y) = 0
This gives us two cases to consider:
Case 1: (x^2 + xy + 4y^2) = 0
Case 2: (x - y) = 0
For Case 1, since the sum of squares is always positive, it is impossible for (x^2 + xy + 4y^2) to be equal to zero. Therefore, we can exclude it.
For Case 2, we have x = y.
Now, let's substitute x = y into our expression E:
E = (x^2 + x^2) / (x^2 + x^2 + 4x^2) = (2x^2) / (6x^2) = 1/3
Thus, the maximum and minimum values of E are both 1/3. Therefore, M = m = 1/3.
To find the average A, we can simply take the average of M and m:
A = (M + m) / 2 = (1/3 + 1/3) / 2 = 2/6 = 1/3.
Therefore, the average value A is 1/3.