Si :x ^ 2 + y ^ 2 + z ^ 2 ≥ xy + yz + zx + K(x - y) ^ 2 , si se cumple para todo numero real , halla el maximo valor de K , la solucion que sea muy detallada ?

<img src="https://tex.z-dn.net/?f=x%5E2%2By%5E2%2Bz%5E2%20%5Cgeq%20xy%20%2Byz%20%2Bzx%2BK%28x-y%29%5E2%5C%5C%20%5C%5C%20%0A%281%29~~%5Ctext%7BSi%20%7Dx%3Dy%5Ctext%7B%20entonces%7D%5C%5C%20%5C%5C%0A2x%5E2%2Bz%5E2%5Cgeq%20x%5E2%20%2B2zx%5C%5C%20%0Ax%5E2%2Bz%5E2%5Cgeq%20%202zx%5C%5C%20%5C%5C%0A%5Ctext%7BSe%20deduce%20que%20K%20puede%20ser%20cualquier%20valor%7D%5C%5C%20%5C%5C%0A%282%29~~%5Ctext%7Bahora%20supongamos%20que%20%7Dx%5Cneq%20y%5C%5C%20%5C%5C%0AK%5Cleq%20%5Cdfrac%7Bx%5E2%2By%5E2%2Bz%5E2%20-%20xy%20-yz%20-xz%7D%7B%28x-y%29%5E2%7D%3D%5Cdfrac%7B%28x-z%29%28y-z%29%7D%7B%28x-y%29%5E2%7D%2B1%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctext%7BSi%20%7Dz%3Dx%5Ctext%7B%20o%20bien%20%7Dz%3Dy%5Ctext%7B%20entonces%20%7DK%5Cleq%201.%20~~%5Ctext%7BHasta%20aqu%5C%27i%20%7DK%5Cin%20%28-%5Cinfty%2C1%5D%20%5C%5C%20%5C%5C%0A" />

<img src="https://tex.z-dn.net/?f=%5Ctext%7BAhora%20toca%20ver%20si%20es%20posible%20acotar%20la%20funci%5C%27on%20%7D%5C%5C%20%5C%5C%0A%5Chspace%2A%7B3cm%7Df%28x%2Cy%2Cz%29%3D%5Cdfrac%7B%28x-z%29%28y-z%29%7D%7B%28x-y%29%5E2%7D%5C%5C%20%5C%5C%20%5C%5C%0A%28a%29~%5Ctext%7BPuntos%20cr%5C%27iticos%7D%5C%5C%20%5C%5C%0Af_x%3D-%5Cdfrac%7B%28x%20%2B%20y%20-%202%20z%29%20%28y%20-%20z%29%7D%7B%28x%20-%20y%29%5E3%7D%3D0%5C%5C%20%5C%5C%20%5C%5C%0Af_y%3D%5Cdfrac%7B%28x%20%2B%20y%20-%202%20z%29%20%28x%20-%20z%29%7D%7B%28x%20-%20y%29%5E3%7D%3D0%5C%5C%20%5C%5C%20%5C%5C%0Af_z%3D-%5Cdfrac%7Bx%20%2B%20y%20-%202%20z%7D%7B%28x%20-%20y%29%5E2%7D%3D0%5C%5C%20%5C%5C%20%5C%5C%0AP%3D%5Cleft%28x%2Cy%2C%5Cdfrac%7Bx%2By%7D%7B2%7D%5Cright%29%0A" />

<img src="https://tex.z-dn.net/?f=%28b%29~%5Ctext%7BClasificaci%5C%27on%20de%20%7DP%3A%20%5C%5C%20%5C%5C%0Af_%7Bxx%7D%28P%29%3D%5Cdfrac%7B1%7D%7B2%28x-y%29%5E2%7D~%2C~f_%7Bxy%7D%3Df_%7Byx%7D%3D%5Cdfrac%7B1%7D%7B2%28x-y%29%5E2%7D~%2C~f_%7Bxz%7D%3Df_%7Bzx%7D%3D-%5Cdfrac%7B1%7D%7B%28x-y%29%5E2%7D%5C%5C%20%5C%5C%20%5C%5C%0Af_%7Byy%7D%28P%29%3D%5Cdfrac%7B1%7D%7B2%28x-y%29%5E2%7D~%2C~f_%7Bzz%7D%3D%5Cdfrac%7B1%7D%7B%28x-y%29%5E2%7D~%2C~f_%7Byz%7D%3Df_%7Bzy%7D%3D-%5Cdfrac%7B1%7D%7B%28x-y%29%5E2%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctext%7BSegunda%20diferencial%20de%20%7Df%5C%5C%20%5C%5C%0AD%5E2f%3Df_%7Bxx%7D%28dx%29%5E2%2Bf_%7Byy%7D%28dy%29%5E2%2Bf_%7Bzz%7D%28dz%29%5E2%2B2%28f_%7Bxy%7Ddx%5C%2Cdy%2Bf_%7Bxz%7Ddx%5C%2Cdz%2B%5Ccdots%5C%5C%20%0A~~~~~~~~~%2Bf_%7Byz%7Ddy%5C%2Cdz%29%0A" />

<img src="https://tex.z-dn.net/?f=%5Ctext%7B...%20evaluando%20en%20P%3A%7D%5C%5C%20%5C%5C%0AD%5E2f%28P%29%3D%5Cdfrac%7B1%7D%7B%28x-y%29%5E2%7D%5Cleft%5B%5Cdfrac%7B%28dx%29%5E2%7D%7B2%7D%2B%5Cdfrac%7B%28dy%29%5E2%7D%7B2%7D%2B2%28dz%29%5E2%2B%5Cdfrac%7Bdx%5C%2Cdy%7D%7B2%7D-dx~dz-dy~dz%5Cright%5D%5C%5C%20%5C%5C%20%5C%5C%0AD%5E2f%28P%29%3D%5Cdfrac%7B1%7D%7B%28x-y%29%5E2%7D%5Cleft%5B%5Cdfrac%7B1%7D%7B4%7D%5Cleft%28dx%2Bdy-2dz%5Cright%29%5E2%2B%5Cdfrac%7B%28dx%29%5E2%7D%7B4%7D%2B%5Cdfrac%7B%28dy%29%5E2%7D%7B4%7D%2B%28dz%29%5E2%5Cright%5D%5C%5C%20%5C%5C%20%5C%5C%0AD%5E2f%28P%29%5C%20%5Ctextgreater%20%5C%200%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctext%7BPor%20ende%20%7DP%5Ctext%7B%20es%20un%20punto%20de%20m%5C%27inimo%20%7D%5C%5C%20%5C%5C%20%0Af%28P%29%3D-%5Cdfrac%7B1%7D%7B4%7D%5CLongrightarrow%20K%5Cleq%20-%5Cdfrac%7B1%7D%7B4%7D%2B1%3D%5Cdfrac%7B3%7D%7B4%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctext%7BPor%20ende%20%7D%5Cboxed%7BK_%7B%5Cmax%7D%3D%5Cdfrac%7B3%7D%7B4%7D%7D" />

Este es el mayor valor de K para que cumpla tal desigualdad para cualesquiera valores de x, y & z.