Valor mínimo de sumatoria?

<img src="https://tex.z-dn.net/?f=%24f%28x%29%3D%5Csum%5Climts_%7Bi%3D1%7D%5En%28x-a_i%29%5E2%24" />

con

<img src="https://tex.z-dn.net/?f=a_j%5C%20%5Ctextless%20%5C%20a_%7Bj%2B1%7D%2C%20%201%5Cleq%20j%20%5Cleq%20n-1" />

Ahora veamos si tiene puntos críticos <img src="https://tex.z-dn.net/?f=%24f%27%28x%29%3D2%5Csum%5Climts_%7Bi%3D1%7D%5En%28x-a_i%29%24" /> <img src="https://tex.z-dn.net/?f=%242%5Csum%5Climts_%7Bi%3D1%7D%5En%28x-a_i%29%3D0%24" /> <img src="https://tex.z-dn.net/?f=%24nx-%5Csum%5Climts_%7Bi%3D1%7D%5En%28a_i%29%3D0%24" /> <img src="https://tex.z-dn.net/?f=%24x%3D%5Cfrac%7B%5Csum%5Climts_%7Bi%3D1%7D%5En%28a_i%29%7D%7Bn%7D%24" />

Es el único punto crítico

(1) si <img src="https://tex.z-dn.net/?f=%24x%3C%5Cfrac%7B%5Csum%5Climts_%7Bi%3D1%7D%5En%28a_i%29%7D%7Bn%7D%24" /> entonces <img src="https://tex.z-dn.net/?f=f%27%28x%29%3C0" />, entonces la función f es decreciente

(2) si <img src="https://tex.z-dn.net/?f=%24x%3E%5Cfrac%7B%5Csum%5Climts_%7Bi%3D1%7D%5En%28a_i%29%7D%7Bn%7D%24" /> \ frac{ \ sum \ limts_{i = 1} ^ n(a_i)}{n}$" alt = "$x> \ frac{ \ sum \ limts_{i = 1} ^ n(a_i)}{n}$" align = "absmiddle" class = "latex - formula"> entonces <img src="https://tex.z-dn.net/?f=f%27%28x%29%3E0" />0" alt = "f'(x)>0" align = "absmiddle" class = "latex - formula">, entonces la función f es creciente

por lo tanto [img = 10] es un mínimo

Entonces deducimos :

[img = 11].