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Classical Inequalities

Your destination for questions of the form "do we know if this expression is always greater than this other expression?" Explore what humans know about mathematical inequalities.

Challenge Quizzes

Level 2

         

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

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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