HB: \( x+2y\) soll extremal werden NB:\(x^2+y^2-1=0\)
\(L(x,y,λ)=x+2y+λ(x^2+y^2-1)\)
1.)\(L_x(x,y,λ)=1+2λx\) 1.) \(1+2λx=0\) 1.) \(2λ=-\frac{1}{x}\)
2.)\(L_y(x,y,λ)=2+2yλ\) 2.) \(2+2yλ=0\) 2.) \(2λ=-\frac{2}{y}\) )
\(-\frac{1}{2x}=-\frac{2}{y}\) → \(\frac{1}{x}=\frac{2}{y}\) →\(y=2x\)
3.) \(L_λ(x,y,λ)=x^2+y^2-1\) → \(x^2+4x^2=1\) → \(x^2=\frac{1}{5}\)
→ \(x_1=\frac{1}{\sqrt{5}}\) \( y_1=\frac{2}{\sqrt{5}}\)
\( x_2=-\frac{1}{\sqrt{5}}\) \( y_2=-\frac{2}{\sqrt{5}}\)
Extremum bei \( \frac{1}{\sqrt{5}}+\frac{4}{\sqrt{5}}=\sqrt{5}\)
und bei \( -\frac{1}{\sqrt{5}}-\frac{4}{\sqrt{5}}=-\sqrt{5}\)
Art der Extrema kann man wie bestimmen?