Question

the value of p for which the sum of the square of the roots of 2x² - 2(p-2)x - p-1 = 0 is

Answer 1

let a, b be the roots of the above equation
 
now a+b = (p-2)
 
ab = -(p+1)
 
we have to find out a^2 + b^2
 
a^2 + b^2 = (a+b)^2 - 2ab = (p-2)^2 + 2(p+1)
 = p^2-4p+4+2p+2 = p^2-2p+6 ----------------> 1
 
differentiate eq 1 wrt p to get 2p-2. equate to 0 to get p=1
 
now the second order derivative is 2 >0. Thus there is a local minima at p=1
 
Thus the sum of squares of the polynomial assumes the least value at p=1