Algebra
# Classical Inequalities

\[\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{d} + \dfrac{d}{a} \]

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

Let \(x\) be a positive real number. Find the minimum value of \[8x^{5}+10x^{-4}.\]

Positive reals \(a,b,c\) satisfy \(abc=1\). Find the minimum value of

\[\dfrac{1+ab}{1+a}+\dfrac{1+bc}{1+b}+\dfrac{1+ca}{1+c}.\]

Enter your answer correct to 3 decimal places.

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

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

Find the minimum value of \(\sqrt{a^2+b^2} \) when \(3a+4b=15\).

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