Apply the **Cauchy-Schwarz Inequality** in the following form:
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- \frac{x^2}{y} + \frac{y^2}{z} + \frac{z^2}{x}.
- Solution: We are to minimize the expression
- subject to $ x + y + z = 1 $, with $ x, y, z > 0 $.
- \left( \frac{x^2}{y} + \frac{y^2}{z} + \frac{z^2}{x} \right)(y + z + x) \geq (x + y + z)^2
- Since $ x + y + z = 1 $, we have:
- \left( \frac{x^2}{y} + \frac{y^2}{z} + \frac{z^2}{x} \right)(1) \geq 1^2 = 1
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