Ayuda porrrrrfiHalle los maximos y mınimos absolutos de : a) f(x, y) = (x − 1)2 + y 2 restringida a la region D = {(x, y) ∈ R 2 / x2 + y 2 ≤ 4}?

1) Sea c ∈ [0, 4] y la restricción <img src="https://tex.z-dn.net/?f=h%28x%2Cy%29%20%3D%20x%5E2%2By%5E2-c%3D0" />2) La función a maximizar es <img src="https://tex.z-dn.net/?f=f%28x%2C%20y%29%20%3D%20%28x-1%29%5E2%20%2B%20y%5E2" />.

Utilicemos la función de Lagrange <img src="https://tex.z-dn.net/?f=%5CLambda%28x%2Cy%29%3D%20%28x-1%29%5E2%20%2B%20y%5E2-%5Clambda%28x%5E2%2By%5E2-c%29%5C%5C%20%5C%5C%0A%5Cbullet%20%5Ctext%7B%20Hallemos%20los%20puntos%20cr%5C%27iticos%20%7D%5C%5C%20%5C%5C%0A%5CLambda_x%3D2%28x-1%29-2%5Clambda%20x%3D0%5C%5C%0A%5CLambda_y%3D2y-2%5Clambda%20y%3D0" /><img src="https://tex.z-dn.net/?f=%28x%2Cy%2C%5Clambda%29%5Cin%5Cleft%5C%7B%5Cleft%28-%5Csqrt%7Bc%7D%2C0%2C%5Cdfrac%7B%5Csqrt%7Bc%7D%2B1%7D%7B%5Csqrt%7Bc%7D%7D%5Cright%29%3B%5Cleft%28%5Csqrt%7Bc%7D%2C0%2C%5Cdfrac%7B%5Csqrt%7Bc%7D-1%7D%7B%5Csqrt%7Bc%7D%7D%5Cright%29%5Cright%5C%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5Cbullet%20%5Ctext%7BCriterio%20de%20la%20segunda%20derivada%3A%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5CLambda_%7Bxx%7D%3D2-2%5Clambda~~%3B~~%20%5CLambda_%7Bxy%7D%20%3D0~~%3B~~%20%5CLambda_%7Byy%7D%20%3D2-2%5Clambda%5C%5C%20%5C%5C%0A%7CM_%7B11%7D%7C%3D2-2%5Clambda%5C%5C%20%5C%5C%0A%7CM_%7B22%7D%7C%3D%5CLambda_%7Bxx%7D%5CLambda_%7Byy%7D-%5CLambda%5E2_%7Bxy%7D%3D%282-2%5Clambda%29%5E2%3E0" />0" alt = "(x, y, \ lambda) \ in \ left \ { \ left( - \ sqrt{c}, 0, \ dfrac{ \ sqrt{c} + 1}{ \ sqrt{c}} \ right) ; \ left( \ sqrt{c}, 0, \ dfrac{ \ sqrt{c} - 1}{ \ sqrt{c}} \ right) \ right \ } \ \ \ \ \ \ &#10 ; \ bullet \ text{Criterio de la segunda derivada : } \ \ \ \ \ \ &#10 ; \ Lambda_{xx} = 2 - 2 \ lambda~~ ; ~~ \ Lambda_{xy} = 0~~ ; ~~ \ Lambda_{yy} = 2 - 2 \ lambda \ \ \ \ &#10 ; |M_{11}| = 2 - 2 \ lambda \ \ \ \ &#10 ; |M_{22}| = \ Lambda_{xx} \ Lambda_{yy} - \ Lambda ^ 2_{xy} = (2 - 2 \ lambda) ^ 2>0" align = "absmiddle" class = "latex - formula"><img src="https://tex.z-dn.net/?f=%5Ctext%7BEntonces%20todo%20depende%20del%20signo%20de%20%24%7CM_%7B11%7D%7C%24%7D%5C%5C%20%5C%5C%0A%5Cbullet%5Cbullet%20%5Ctext%7B%20Para%20el%20punto%20%7D%20%5Cleft%28-%5Csqrt%7Bc%7D%2C0%2C%5Cdfrac%7B%5Csqrt%7Bc%7D%2B1%7D%7B%5Csqrt%7Bc%7D%7D%5Cright%29%3A%5C%5C%20%5C%5C%20%5C%5C%0A%7CM_%7B11%7D%7C%3D2-2%5Ctimes%5Cdfrac%7B%5Csqrt%7Bc%7D%2B1%7D%7B%5Csqrt%7Bc%7D%7D%3D-%5Cdfrac%7B2%7D%7B%5Csqrt%7Bc%7D%7D%3C0%5C%5C%20%5C%5C%20%5C%5C%20%5C%5C%0A%5Cbullet%5Cbullet%20%5Ctext%7B%20Para%20el%20punto%20%7D%20%5Cleft%28%5Csqrt%7Bc%7D%2C0%2C%5Cdfrac%7B%5Csqrt%7Bc%7D-1%7D%7B%5Csqrt%7Bc%7D%7D%5Cright%29%3A%5C%5C%20%5C%5C%20%5C%5C%0A%7CM_%7B11%7D%7C%3D2-2%5Ctimes%5Cdfrac%7B%5Csqrt%7Bc%7D-1%7D%7B%5Csqrt%7Bc%7D%7D%3D%5Cdfrac%7B2%7D%7B%5Csqrt%7Bc%7D%7D%3E0" />

<img src="https://tex.z-dn.net/?f=%5Ctext%7BPor%20ende%20%24%28-%5Csqrt%7Bc%7D%2C0%29%24%20es%20un%20punto%20de%20m%5C%27aximo%20y%20%24%28%5Csqrt%7Bc%7D%2C0%29%24%20de%20m%5C%27inimo%7D%5C%5C%20%5C%5C%20%5C%5C%0Af%28-%5Csqrt%7Bc%7D%2C0%29%3D%28%5Csqrt%7Bc%7D%2B1%29%5E2%5Cto%20f_%7B%5Cmax%7D%3D9~~%2C~~%5Ctext%7BCuando%20%24c%3D4%24%7D%5C%5C%20%5C%5C%0Af%28%5Csqrt%7Bc%7D%2C0%29%3D%28%5Csqrt%7Bc%7D-1%29%5E2%5Cto%20f_%7B%5Cmin%7D%3D0~~%2C~~%5Ctext%7BCuando%20%24c%3D1%24%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctext%7BAs%5C%27i%20%24%28-2%2C0%29%24%20es%20un%20punto%20de%20m%5C%27aximo%20y%20%24%281%2C0%29%24%20uno%20de%20m%5C%27inimo%7D" />.