harder 3U: inequalities (1 Viewer)

[jkwii]

how do you do this one? last Q in patel.

for x, y, z, a, b, c are real numbers prove that:

(x^2 + y^2 + z^2)(a^2 + b^2 + c^2) > (or equal to) (ax + by + cz)^2

 

[3unitz]

cauchy schwarz inequality

http://www.artofproblemsolving.com/Wiki/index.php/Cauchy-Schwarz_Inequality

 

[jkwii]

can u explain it properly, the theorem is hard to understand from the symbols... not used to reading and learning off a wiki article.

 

[3unitz]

can u explain it properly, the theorem is hard to understand from the symbols... not used to reading and learning off a wiki article.

consider the quadratic:

y = (ax + b)^2 + (cx + d)^2 + (ex + f)^2 (a, b, c, d, e, f are real)

since y >= 0, it follows that the quadratic either has 1 or no real roots.

this implies the discriminant <= 0.

(2ab + 2cd + 2ef)^2 - 4(a^2 + c^2 + e^2)(b^2 + d^2 + f^2) <= 0

(ab + cd + ef)^2 <= (a^2 + c^2 + e^2)(b^2 + d^2 + f^2)