La ley de hardy - weinberg - Usar multiplicadores deLaGrange para hacer máximo el valor deP(p, q, r) = 2pq +2pr + 2qr sujeto a p + q + r = 1?

<img src="https://tex.z-dn.net/?f=L%3D2pq%2B2pr%2B2qr-%5Clambda%28p%2Bq%2Br-1%29%0A%0A" />

Hallemos las primeras derivadas

<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%0A%5Cfrac%7B%5Cpartial%20L%7D%7B%5Cpartial%20p%7D%3D2q%2B2r-%5Clambda%5C%5C%20%5C%5C%0A%5Cfrac%7B%5Cpartial%20L%7D%7B%5Cpartial%20q%7D%3D2p%2B2r-%5Clambda%5C%5C%20%5C%5C%0A%5Cfrac%7B%5Cpartial%20L%7D%7B%5Cpartial%20r%7D%3D2p%2B2q-%5Clambda%5C%5C%20%5C%5C" />

Hallemos los puntos estacionarios

<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%0A2q%2B2r-%5Clambda%3D0%5C%5C%20%5C%5C%0A2p%2B2r-%5Clambda%3D0%5C%5C%20%5C%5C%0A2p%2B2q-%5Clambda%3D0%5C%5C%20%5C%5C%0A%5Ctext%7Bsumemos%3A%7D%5C%5C%0A4%28p%2Bq%2Br%29-3%5Clambda%20%3D0%5C%5C%20%5C%5C%0A%5Cboxed%7B%5Clambda%20%3D%20%5Cfrac%7B4%7D%7B3%7D%7D%20%5C%5C%20%5C%5C%0A%5Cboxed%7Bp%3Dq%3Dr%3D%5Cfrac%7B1%7D%7B3%7D%7D" />

entonces el posible extremos es <img src="https://tex.z-dn.net/?f=%24%28p%2Cq%2Cr%29%3D%5Cleft%28%5Cfrac%7B1%7D%7B3%7D%3B%5Cfrac%7B1%7D%7B3%7D%3B%5Cfrac%7B1%7D%7B3%7D%5Cright%29%24" />

Ahora hallemos las segundas derivadas

<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%0A%5Cfrac%7B%5Cpartial%5E2%20L%7D%7B%5Cpartial%20p%5E2%7D%3D0%5C%5C%20%5C%5C%0A%5Cfrac%7B%5Cpartial%5E2%20L%7D%7B%5Cpartial%20p%5C%2C%5Cpartial%20q%7D%3D2%3D%5Cfrac%7B%5Cpartial%5E2%20L%7D%7B%5Cpartial%20q%5C%2C%5Cpartial%20p%7D%5C%5C%20%5C%5C%0A%5Cfrac%7B%5Cpartial%5E2%20L%7D%7B%5Cpartial%20q%5E2%7D%3D0" />

<img src="https://tex.z-dn.net/?f=L_%7Brr%7D%3D0%5C%5C%20%5C%5C%0AL_%7Bpr%7D%3DL_%7Bqr%7D%3D2" />

Si nos fijamos en<img src="https://tex.z-dn.net/?f=L_%7Bpp%7DL_%7Bqq%7D-L_%7Bpq%7D%5E2%3D-4" /> es negativa, por ello no hay extremos.