Determine os valores de máximos, mínimos e ponto de sela: f(x,y) = 2x^3+xy²+5x²+y²
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Encontramos esses pontos quando igualamos derivada parcial de x e derivada parcial de y a 0:
![f_x=6x^2+y^2+10x\\\\f_y=2xy+2y\\\\\\ \left \{ {{6x^2+y^2+10x=0} \atop {2xy+2y=0}} \right. \\\\\\ * 2xy+2y=0\\\\2y(x+1)=0\\\\y=0,\ x=-1\\\\\\ * 6x^2+y^2+10x=0\\\\6(-1)^2+y^2+10(-1)=0\\\\y^2=4\\\\y=\pm2\\\\6x^2+(0)^2+10x=0\\\\x=0,\ x=-\dfrac53\\\\\\S:\left\{(-1,2),(-1,-2),(0,0),\left(0,-\dfrac53\right)\right\} f_x=6x^2+y^2+10x\\\\f_y=2xy+2y\\\\\\ \left \{ {{6x^2+y^2+10x=0} \atop {2xy+2y=0}} \right. \\\\\\ * 2xy+2y=0\\\\2y(x+1)=0\\\\y=0,\ x=-1\\\\\\ * 6x^2+y^2+10x=0\\\\6(-1)^2+y^2+10(-1)=0\\\\y^2=4\\\\y=\pm2\\\\6x^2+(0)^2+10x=0\\\\x=0,\ x=-\dfrac53\\\\\\S:\left\{(-1,2),(-1,-2),(0,0),\left(0,-\dfrac53\right)\right\}](https://tex.z-dn.net/?f=f_x%3D6x%5E2%2By%5E2%2B10x%5C%5C%5C%5Cf_y%3D2xy%2B2y%5C%5C%5C%5C%5C%5C+%5Cleft+%5C%7B+%7B%7B6x%5E2%2By%5E2%2B10x%3D0%7D+%5Catop+%7B2xy%2B2y%3D0%7D%7D+%5Cright.+%5C%5C%5C%5C%5C%5C+%2A+2xy%2B2y%3D0%5C%5C%5C%5C2y%28x%2B1%29%3D0%5C%5C%5C%5Cy%3D0%2C%5C+x%3D-1%5C%5C%5C%5C%5C%5C+%2A+6x%5E2%2By%5E2%2B10x%3D0%5C%5C%5C%5C6%28-1%29%5E2%2By%5E2%2B10%28-1%29%3D0%5C%5C%5C%5Cy%5E2%3D4%5C%5C%5C%5Cy%3D%5Cpm2%5C%5C%5C%5C6x%5E2%2B%280%29%5E2%2B10x%3D0%5C%5C%5C%5Cx%3D0%2C%5C+x%3D-%5Cdfrac53%5C%5C%5C%5C%5C%5CS%3A%5Cleft%5C%7B%28-1%2C2%29%2C%28-1%2C-2%29%2C%280%2C0%29%2C%5Cleft%280%2C-%5Cdfrac53%5Cright%29%5Cright%5C%7D)
Agora que temos os pontos podemos determinar quem eh o que com o teste da segunda derivada:
![\text{max:}\\\\f_{xx}\ \textless \ 0\text{ and }(f_{xx})(f_{yy})-(f_{xy})^2\ \textgreater \ 0\\\\\\\text{min:}\\\\f_{xx}\ \textgreater \ 0\text{ and }(f_{xx})(f_{yy})-(f_{xy})^2\ \textgreater \ 0\\\\\\\text{saddle:}\\\\(f_{xx})(f_{yy})-(f_{xy})^2\ \textless \ 0\\\\\\f_{xx}=12x+10\\\\f_{yy}=2x+2\\\\f_{xy}=2y\\\\\\(-1,2):\\\\f_{xx}=12(-1)+10=-2<0\\\\f_{yy}=2(-1)+2=0\\\\f_{xy}=2(2)=4\\\\(-2)(0)-(4)^2=-16<0\text{ (saddle)}\\\\\\(-1,-2):\\\\f_{xx}=12(-1)+10=-2<0\\\\f_{yy}=2(-1)+2=0\\\\f_{xy}=2(-2)=-4\\\\(-2)(0)-(-4)^2=-16<0\text{ (saddle)}\\\\\\(0,0):\\\\f_{xx}=12(0)+10=10>0\\\\f_{yy}=2(0)+2=2\\\\f_{xy}=2(0)=0\\\\(10)(2)-(0)^2=20>0\text{ (min)}\\\\\\\left(0,-\dfrac53\right)\\\\f_{xx}=12(0)+10=10>0\\\\f_{yy}=2(0)+2=2\\\\f_{xy}=2\left(-\dfrac53\right)=-\dfrac{10}{3}\\\\(10)(2)-\left(-\dfrac{10}{3}\right)^2=\dfrac{80}{9}>0\text{ (min)} \text{max:}\\\\f_{xx}\ \textless \ 0\text{ and }(f_{xx})(f_{yy})-(f_{xy})^2\ \textgreater \ 0\\\\\\\text{min:}\\\\f_{xx}\ \textgreater \ 0\text{ and }(f_{xx})(f_{yy})-(f_{xy})^2\ \textgreater \ 0\\\\\\\text{saddle:}\\\\(f_{xx})(f_{yy})-(f_{xy})^2\ \textless \ 0\\\\\\f_{xx}=12x+10\\\\f_{yy}=2x+2\\\\f_{xy}=2y\\\\\\(-1,2):\\\\f_{xx}=12(-1)+10=-2<0\\\\f_{yy}=2(-1)+2=0\\\\f_{xy}=2(2)=4\\\\(-2)(0)-(4)^2=-16<0\text{ (saddle)}\\\\\\(-1,-2):\\\\f_{xx}=12(-1)+10=-2<0\\\\f_{yy}=2(-1)+2=0\\\\f_{xy}=2(-2)=-4\\\\(-2)(0)-(-4)^2=-16<0\text{ (saddle)}\\\\\\(0,0):\\\\f_{xx}=12(0)+10=10>0\\\\f_{yy}=2(0)+2=2\\\\f_{xy}=2(0)=0\\\\(10)(2)-(0)^2=20>0\text{ (min)}\\\\\\\left(0,-\dfrac53\right)\\\\f_{xx}=12(0)+10=10>0\\\\f_{yy}=2(0)+2=2\\\\f_{xy}=2\left(-\dfrac53\right)=-\dfrac{10}{3}\\\\(10)(2)-\left(-\dfrac{10}{3}\right)^2=\dfrac{80}{9}>0\text{ (min)}](https://tex.z-dn.net/?f=+%5Ctext%7Bmax%3A%7D%5C%5C%5C%5Cf_%7Bxx%7D%5C+%5Ctextless+%5C+0%5Ctext%7B+and+%7D%28f_%7Bxx%7D%29%28f_%7Byy%7D%29-%28f_%7Bxy%7D%29%5E2%5C+%5Ctextgreater+%5C+0%5C%5C%5C%5C%5C%5C%5Ctext%7Bmin%3A%7D%5C%5C%5C%5Cf_%7Bxx%7D%5C+%5Ctextgreater+%5C+0%5Ctext%7B+and+%7D%28f_%7Bxx%7D%29%28f_%7Byy%7D%29-%28f_%7Bxy%7D%29%5E2%5C+%5Ctextgreater+%5C+0%5C%5C%5C%5C%5C%5C%5Ctext%7Bsaddle%3A%7D%5C%5C%5C%5C%28f_%7Bxx%7D%29%28f_%7Byy%7D%29-%28f_%7Bxy%7D%29%5E2%5C+%5Ctextless+%5C+0%5C%5C%5C%5C%5C%5Cf_%7Bxx%7D%3D12x%2B10%5C%5C%5C%5Cf_%7Byy%7D%3D2x%2B2%5C%5C%5C%5Cf_%7Bxy%7D%3D2y%5C%5C%5C%5C%5C%5C%28-1%2C2%29%3A%5C%5C%5C%5Cf_%7Bxx%7D%3D12%28-1%29%2B10%3D-2%26lt%3B0%5C%5C%5C%5Cf_%7Byy%7D%3D2%28-1%29%2B2%3D0%5C%5C%5C%5Cf_%7Bxy%7D%3D2%282%29%3D4%5C%5C%5C%5C%28-2%29%280%29-%284%29%5E2%3D-16%26lt%3B0%5Ctext%7B+%28saddle%29%7D%5C%5C%5C%5C%5C%5C%28-1%2C-2%29%3A%5C%5C%5C%5Cf_%7Bxx%7D%3D12%28-1%29%2B10%3D-2%26lt%3B0%5C%5C%5C%5Cf_%7Byy%7D%3D2%28-1%29%2B2%3D0%5C%5C%5C%5Cf_%7Bxy%7D%3D2%28-2%29%3D-4%5C%5C%5C%5C%28-2%29%280%29-%28-4%29%5E2%3D-16%26lt%3B0%5Ctext%7B+%28saddle%29%7D%5C%5C%5C%5C%5C%5C%280%2C0%29%3A%5C%5C%5C%5Cf_%7Bxx%7D%3D12%280%29%2B10%3D10%26gt%3B0%5C%5C%5C%5Cf_%7Byy%7D%3D2%280%29%2B2%3D2%5C%5C%5C%5Cf_%7Bxy%7D%3D2%280%29%3D0%5C%5C%5C%5C%2810%29%282%29-%280%29%5E2%3D20%26gt%3B0%5Ctext%7B+%28min%29%7D%5C%5C%5C%5C%5C%5C%5Cleft%280%2C-%5Cdfrac53%5Cright%29%5C%5C%5C%5Cf_%7Bxx%7D%3D12%280%29%2B10%3D10%26gt%3B0%5C%5C%5C%5Cf_%7Byy%7D%3D2%280%29%2B2%3D2%5C%5C%5C%5Cf_%7Bxy%7D%3D2%5Cleft%28-%5Cdfrac53%5Cright%29%3D-%5Cdfrac%7B10%7D%7B3%7D%5C%5C%5C%5C%2810%29%282%29-%5Cleft%28-%5Cdfrac%7B10%7D%7B3%7D%5Cright%29%5E2%3D%5Cdfrac%7B80%7D%7B9%7D%26gt%3B0%5Ctext%7B+%28min%29%7D)
Agora que temos os pontos podemos determinar quem eh o que com o teste da segunda derivada:
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