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[tex]f(x,y,z)=\log(x)+\log(y)+3\log(z)[/tex]
[tex]\nabla f(x,y,z)=(\frac{1}{x},\frac{1}{x},\frac{3}{x})[/tex]
[tex]\nabla g(x,y,z)=(2x,2y,2z)[/tex]
[tex]1/x=2x\lambda[/tex]
[tex]1/y=2y\lambda[/tex]
[tex]3/z=2z\lambda[/tex]
[tex]x^2+y^2+z^2=5r^2[/tex]
[tex]1/(2\lambda)=x^2, \space 1/(2\lambda)=y^2, \space 3/(2\lambda)=x^2[/tex]
[tex]1/(2\lambda)+1/(2\lambda)+3/(2\lambda)=5r^2 \rightarrow \space \lambda= \frac{1}{2r^2}[/tex]
[tex]x=r, \space y=r, \space z=\sqrt{3}r[/tex]
[tex]f(r,r,\sqrt{3r})_{max}= 3\log(r)+ \frac{\log(3)}{2}[/tex]
Resten burde vel være greit, bør være mulig å se noen likheter mellom f(x,y,z) og ulikheten under
[tex]\nabla f(x,y,z)=(\frac{1}{x},\frac{1}{x},\frac{3}{x})[/tex]
[tex]\nabla g(x,y,z)=(2x,2y,2z)[/tex]
[tex]1/x=2x\lambda[/tex]
[tex]1/y=2y\lambda[/tex]
[tex]3/z=2z\lambda[/tex]
[tex]x^2+y^2+z^2=5r^2[/tex]
[tex]1/(2\lambda)=x^2, \space 1/(2\lambda)=y^2, \space 3/(2\lambda)=x^2[/tex]
[tex]1/(2\lambda)+1/(2\lambda)+3/(2\lambda)=5r^2 \rightarrow \space \lambda= \frac{1}{2r^2}[/tex]
[tex]x=r, \space y=r, \space z=\sqrt{3}r[/tex]
[tex]f(r,r,\sqrt{3r})_{max}= 3\log(r)+ \frac{\log(3)}{2}[/tex]
Resten burde vel være greit, bør være mulig å se noen likheter mellom f(x,y,z) og ulikheten under