This problem is solved by finding the maximum and minimum of the function
f(p,q,r) = pq + qr + rp using Lagrange Multiplier, subject to the constraint p^2 + q^2 + r^2 =1. This means, the point (p,q,r) is on the surface of the sphere with radius 1 and centered at the origin. Define
H(p.q,r,l) = f(p,q,r) + l(p^2 + q^2 + r^2 – 1). Let the partial derivatives of H with respect to p, q, r and l be all equal to zero, and solve for p, q, and r. We will get + or - sqrt(3) as values of each of p, q and r. When the values of these variables have the same sign, we get the maximum value 1 of f(p,q,r). If any two of the values have different signs, we get the minimum value, -1/3. Therefore, -1/3 < or = pq + qr + rp < or = 1.
Answered On 08/16/2017 11:44