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# Power Mean Inequalities

This chain of inequalities forms the foundation for many other classical inequalities. See how the four common "means" - arithmetic, geometric, harmonic, and quadratic - relate to each other.

\[\large \dfrac{x^2+2-\sqrt{x^4+4}}{x}\]
For real \(x\), find the closed form of the maximum value of the expression above.

Give your answer to 3 decimal places.

**Note:** Try to solve this problem without using calculus.

\(a\), \(b\), and \(c\) are real numbers such that

\[a+b=6 \\ a^2+b^2+c^2=18\]

What is the value of \(a-c\)?

\[ \large (1 + a)(1 + b)(1 + c)(1 + d) \]

If \(a,b,c\) and \(d\) are four positive real numbers such that \(a\times b\times c\times d = 1\), then find the minimum value of the expression above.

For \(a,b,c>0\) and \(a+b+c=6\). Find the minimum value of

\[ \large \left (a+\frac{1}{b} \right )^{2}+ \left (b+\frac{1}{c} \right )^{2}+\left (c+\frac{1}{a} \right )^{2} \]

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