1. Prove that for all real x,y the value of x2+2xy+3y2-6x-2y cannot be less than -11.
More I'll Post later.
11To prove that
x2+2xy+3y2-6x-2y (greater than equal to) -11
or x2+2x(y-3)+ 3y2-2y+11(greater than equal to) 0 for all real values of x.
If roots of corresponding equations are imaginary or real and equal,i.e,
if 4(y-3)2- 4(3y2-2y+11) <= 0
or (y-3)2-(3y2-2y+11)<= 0
or (y2-6y+9)-(3y2-2y+11)<= 0
or -2y2- 4y- 2<= 0
or y2+2y+1>= 0
which is true for all real y, since roots of corresponding equation y2+2y+1 =0 are real and equal.
71Nice.. But you did it by yourself only na?
Some more:
1. Suppose p is the first term of n (n>1) A.M's between two positive nos. a and b ; q be the first of n H.M's between the same two nos, then -->
A.) The value of p:
B) The value of q:
C) If p = 8 and n= 3, then
i) q lies between 8 and 32
ii) q lies between 8 and 16
iii) q doesn't lies between 8 and 32
iv) q does not lie between 8 and 16