Suppose we have the following identity: #(px + (1-p)y)^2 = Ax^2 + Bxy + Cy^2.# Find the minimum of #max(A,B,C)# over #0 leq p leq 1#?
1 Answer
The minimum value of
Explanation:
Expand the right-hand side:
Equate coefficients:
Let us find when any of the two variables intersect (as this will determine when
Substituting arbitrary values in the ranges
When
Thus, the minimum value of
This can be shown with a graph:
graph{(y-x^2)(y-(1-x)^2)(y-2x(1-x))=0 [-0.1, 1.1, -0.1, 1.1]}