Bah calculus calculus & more calculus

[mathstruggle]

got a simple problem.
the question is find min value for f=x*y+(z^2) with constraints 2*x -y=8, and co-ordinates where it occurs.
so far what i did.
▽f=λ*▽g
F(x,y,z)=f(x,y)-λg(x,y,z)
F=(xy+z^2)-λ(2x-y-8)

y=2λ
x=λ

i got λ=-2

is this right? and is that the only λ value?

how do i even start to find equilibrium solutions & general solution to ((t/2)-2)*sin(y), we weren't taught or shown how to find it involving sin,cos, tan?
for general solution i should take it as separable to find the general solution? but how do i start? sin, cos, tan bah

 

[lanedance]

you could probaby substitute directly into that function & minimise

 

[lanedance]

otherwise lagrange multipliers are alway good...

 

[HallsofIvy]

got a simple problem.
the question is find min value for f=x*y+(z^2) with constraints 2*x -y=8, and co-ordinates where it occurs.
so far what i did.
▽f=λ*▽g
F(x,y,z)=f(x,y)-λg(x,y,z)
F=(xy+z^2)-λ(2x-y-8)

y=2λ
x=λ

i got λ=-2

is this right? and is that the only λ value?

I think you are trying to use "Lagrange multipliers" as lanedance suggested but you seem to have no idea how to do that.

You write, correctly, that $\nabla f= \lambda \nabla g$ but then "F(x,y,z)=f(x,y)-λg(x,y,z)" and "F=(xy+z^2)-λ(2x-y-8)" which have nothing to do with what you wrote previously.

$\nabla f$ is the vector $y\vec{i}+ x\vec{j}+ 2z\vec{k}$ and $\nabla g= 2\vec{i}- \vec{j}$.

"$\nabla f= \lambda \nabla g$" is now
$y\vec{i}+ x\vec{j}+ 2z\vec{k}= \lambda 2\vec{i}- \vec{j}$

Looking at the individual components of that, $y= 2\lambda$, $x= -\lambda$, and $z= 0$.

Now, you have $y= 2\lambda$ but you have [itex]x= \lambda[\itex] rather than $x= -\lambda$. Perhaps that is just a typo. In any case, they do <b>not</b> give "$\lambda= -2$ because you have no reason to believe x= -2 or y= -4!.<br /> <br /> Rather, $x= -\lambda$ says that $\lambda= -x$ and so $y= 2\lambda= -2x$. Putting that into the constraint 2x- y= 8 gives 2x+ 2x= 4x= 8 so x= 2 and y= -4. The solution is (2, -4, 0).<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> how do i even start to find equilibrium solutions & general solution to ((t/2)-2)*sin(y), we weren't taught or shown how to find it involving sin,cos, tan? <br /> for general solution i should take it as separable to find the general solution? but how do i start? sin, cos, tan bah </div> </div> </blockquote> Is this a completely different problem? Then <b>what</b> is the problem? I associate "equilibrium solutions & general solution" with differential equations but you give no differential equation. Are you talking about dy/dt= ((t/2)- 2)sin(y) or some other problem? Please state the entire problem.<br /> <br /> And a general suggestion: if you want people who are really good at math to help you (hopefully more politely than I did), stop dissing mathematics![/itex]

 

[mathstruggle]

I think you are trying to use "Lagrange multipliers" as lanedance suggested but you seem to have no idea how to do that.

You write, correctly, that $\nabla f= \lambda \nabla g$ but then "F(x,y,z)=f(x,y)-λg(x,y,z)" and "F=(xy+z^2)-λ(2x-y-8)" which have nothing to do with what you wrote previously.

$\nabla f$ is the vector $y\vec{i}+ x\vec{j}+ 2z\vec{k}$ and $\nabla g= 2\vec{i}- \vec{j}$.

"$\nabla f= \lambda \nabla g$" is now
$y\vec{i}+ x\vec{j}+ 2z\vec{k}= \lambda 2\vec{i}- \vec{j}$

Looking at the individual components of that, $y= 2\lambda$, $x= -\lambda$, and $z= 0$.

Now, you have $y= 2\lambda$ but you have [itex]x= \lambda[\itex] rather than $x= -\lambda$. Perhaps that is just a typo. In any case, they do <b>not</b> give "$\lambda= -2$ because you have no reason to believe x= -2 or y= -4!.<br /> <br /> Rather, $x= -\lambda$ says that $\lambda= -x$ and so $y= 2\lambda= -2x$. Putting that into the constraint 2x- y= 8 gives 2x+ 2x= 4x= 8 so x= 2 and y= -4. The solution is (2, -4, 0). Is this a completely different problem? Then <b>what</b> is the problem? I associate "equilibrium solutions & general solution" with differential equations but you give no differential equation. Are you talking about dy/dt= ((t/2)- 2)sin(y) or some other problem? Please state the entire problem.<br /> <br /> And a general suggestion: if you want people who are really good at math to help you (hopefully more politely than I did), stop dissing mathematics![/itex]

$i didn't diss maths. why would i diss maths if I am majoring in maths? its called sense of humour. last time I am using this forum, its like dictatorship here no freedom.$