practice problems pending

Q1) (a,b,c) are real numbers such that

a+b+c=0

a^2+b^2+c^2=1

Show that

a^2b^2c^2 <= 1/54.

S1)
Let's construct a cubic equation with roots a,b,c.
(a+b+c)^2 = 1 + 2(ab+bc+ca) => ab + bc + ca = -1/2

x^3 -x/2 - r = 0 where r = abc
Since all 3 roots are real, the product of values of function at critical points(maxima,minima) should be <= 0.
Find critical points, derivative = 3x^2 - 1/2 = 0 => x = -+1/sqrt(6)
f(1/rt(6)) = 1/6.rt(6) - 1/2.rt(6) - r = 1/3.rt(6) - r
f(-1/rt(6) = -1/3.rt(6) - r
multiply:
r^2 - 1/54 <= 0
H.P.

Q2.

Text from the image:

P2) If
1/x + 1/y + 1/z = 1 for x,y,z > 0
pr. th. (x-1)(y-1)(z-1) >= 8

S2.
let a = 1/x and similarly others.
a + b + c = 1
x-1 = 1/a - 1 = (1-a)/a = (b+c)/a
b+c >= 2.rt(bc)
So expr. becomes:
(b+c)(c+a)(a+b)/abc>= 8.abc/abc = 8
H.P.