Find the smallest possible value of
\(\frac{(y-x)^2}{(y-z)(z-x)} + \frac{(z-y)^2}{(z-x)(x-y)} + \frac{(x-z)^2}{(x-y)(y-z)}\)
where x,y and z are distinct real numbers.
+26367 Find the smallest possible value of
\(\dfrac{(y-x)^2}{(y-z)(z-x)} + \dfrac{(z-y)^2}{(z-x)(x-y)} + \dfrac{(x-z)^2}{(x-y)(y-z)} \)
where x,y and z are distinct real numbers.
\(\huge {AM \geq GM} \)
\(\begin{array}{|rcll|} \hline \dfrac{\dfrac{(y-x)^2}{(y-z)(z-x)} + \dfrac{(z-y)^2}{(z-x)(x-y)} + \dfrac{(x-z)^2}{(x-y)(y-z)} }{3} &\ge& \sqrt[3]{\dfrac{(y-x)^2}{(y-z)(z-x)} \times \dfrac{(z-y)^2}{(z-x)(x-y)} \times \dfrac{(x-z)^2}{(x-y)(y-z)}} \\ \dfrac{\dfrac{(y-x)^2}{(y-z)(z-x)} + \dfrac{(z-y)^2}{(z-x)(x-y)} + \dfrac{(x-z)^2}{(x-y)(y-z)} }{3} &\ge& \sqrt[3]{ \dfrac{(y-x)^2(z-y)^2(x-z)^2} {(y-x)^2(z-y)^2(x-z)^2} } \\ \dfrac{\dfrac{(y-x)^2}{(y-z)(z-x)} + \dfrac{(z-y)^2}{(z-x)(x-y)} + \dfrac{(x-z)^2}{(x-y)(y-z)} }{3} &\ge& \sqrt[3]{1} \\ \dfrac{\dfrac{(y-x)^2}{(y-z)(z-x)} + \dfrac{(z-y)^2}{(z-x)(x-y)} + \dfrac{(x-z)^2}{(x-y)(y-z)} }{3} &\ge& 1 \\ \dfrac{(y-x)^2}{(y-z)(z-x)} + \dfrac{(z-y)^2}{(z-x)(x-y)} + \dfrac{(x-z)^2}{(x-y)(y-z)} &\ge& 1\times 3 \\ \dfrac{(y-x)^2}{(y-z)(z-x)} + \dfrac{(z-y)^2}{(z-x)(x-y)} + \dfrac{(x-z)^2}{(x-y)(y-z)} &\ge& 3 \\ \hline \end{array} \)
The smallest possible value is \(\mathbf{3}\)
