Minimize the Expression Five

Algebra Level 5

If $a, b, c, d > 0$ and $c^2 + d^2 = (a^2 + b^2)^3 ,$

then find the greatest lower bound of $\frac{a^3}{c} + \frac{b^3}{d}.$

1 solution

Alan Yan
Sep 12, 2015

We know that $(a^2+b^2)^{\frac{3}{2}} = \sqrt{c^2+d^2}$

By Cauchy-Swartz,

$(\frac{a^3}{c} + \frac{b^3}{d})(ac + bd) \geq (a^2 + b^2)^2 \geq \sqrt{(a^2+b^2)(c^2+d^2)} \geq ac + bd$

Therefore, this implies that $\frac{a^3}{c} + \frac{b^3}{d} \geq \boxed{1}$ .

Alternate Solution:

By Holder, $\left(\frac{a^3}{c} + \frac{b^3}{d}\right)^2(c^2 + d^2) \geq (a^2 + b^2)^3 = (c^2 + d^2)$ and the result follows.