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adm_strat
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[SOLVED] Legrange-->System of equations
Suppose (a,b) is on the graph of [tex]f(x)=x^{2}[/tex] and (c,d) is on the graph of g(x)=ln(x)
a) Accurately approximate the minimum distance between (a,b) and (c,d)
b) Accurately approximate (a,b) and (c,d)
c) What is the relationship between f'(a) and g'(c)
Just a Legrange
The function that needs to be minimized is [tex]\sqrt{(a-c)^{2} + (b-d)^{2}}[/tex]
Which can be left as [tex](a-c)^{2} + (b-d)^{2}[/tex] because they will both be minimized at the same place
The Legrange:
[tex]L(a,b,c,d,\lambda ,\mu )=(a-c)^{2} + (b-d)^{2} - \lambda (a - b^{2}) - \mu (b-ln(d))[/tex]
This gave me the six equations:
[tex]L_{a}= 2a - 2c - \lambda = 0[/tex]
[tex]L_{b}= 2b - 2d - 2\lambda b = 0[/tex]
[tex]L_{c}= 2c - 2a - \mu = 0[/tex]
[tex]L_{d}= 2d - 2b - \frac{\mu }{d}= 0[/tex]
[tex]L_{\lambda }= a = b^{b}[/tex]
[tex]L_{\mu }= c = ln (d) [/tex]
This is where I am stuck
I tried for quite some time to solve for the system of equations and I am at a point where i don't know where to go.
The things I got out of them is:
[tex]\lambda[/tex] = -[tex]\mu[/tex] This is from [tex]L_{a}=L_{c}[/tex]
and
[tex]bd=\frac{1}{2}[/tex]
I got this by taking [tex]L_{a}=0[/tex] and [tex]L_{b}=0[/tex] therefore:
[tex]L_{a} = L_{b} [/tex] --> [tex]L_{a} - L_{b} = 0 [/tex] --> [tex]L_{a} - L_{b} = L_{c} [/tex]
And so on till I got: [tex]L_{a} - L_{b} - L_{c} - L_{d} = 0 [/tex]
Then using some substitution for [tex]\lambda[/tex] and [tex]\mu[/tex] I got bd =[tex]\frac{1}{2}[/tex]
The instructor of my course said I could use software on any of the problems in the handout but even when I tried plugging the equations into mathematica I got an error. I just don't know where to go with the equations.
Homework Statement
Suppose (a,b) is on the graph of [tex]f(x)=x^{2}[/tex] and (c,d) is on the graph of g(x)=ln(x)
a) Accurately approximate the minimum distance between (a,b) and (c,d)
b) Accurately approximate (a,b) and (c,d)
c) What is the relationship between f'(a) and g'(c)
Homework Equations
Just a Legrange
The Attempt at a Solution
The function that needs to be minimized is [tex]\sqrt{(a-c)^{2} + (b-d)^{2}}[/tex]
Which can be left as [tex](a-c)^{2} + (b-d)^{2}[/tex] because they will both be minimized at the same place
The Legrange:
[tex]L(a,b,c,d,\lambda ,\mu )=(a-c)^{2} + (b-d)^{2} - \lambda (a - b^{2}) - \mu (b-ln(d))[/tex]
This gave me the six equations:
[tex]L_{a}= 2a - 2c - \lambda = 0[/tex]
[tex]L_{b}= 2b - 2d - 2\lambda b = 0[/tex]
[tex]L_{c}= 2c - 2a - \mu = 0[/tex]
[tex]L_{d}= 2d - 2b - \frac{\mu }{d}= 0[/tex]
[tex]L_{\lambda }= a = b^{b}[/tex]
[tex]L_{\mu }= c = ln (d) [/tex]
This is where I am stuck
I tried for quite some time to solve for the system of equations and I am at a point where i don't know where to go.
The things I got out of them is:
[tex]\lambda[/tex] = -[tex]\mu[/tex] This is from [tex]L_{a}=L_{c}[/tex]
and
[tex]bd=\frac{1}{2}[/tex]
I got this by taking [tex]L_{a}=0[/tex] and [tex]L_{b}=0[/tex] therefore:
[tex]L_{a} = L_{b} [/tex] --> [tex]L_{a} - L_{b} = 0 [/tex] --> [tex]L_{a} - L_{b} = L_{c} [/tex]
And so on till I got: [tex]L_{a} - L_{b} - L_{c} - L_{d} = 0 [/tex]
Then using some substitution for [tex]\lambda[/tex] and [tex]\mu[/tex] I got bd =[tex]\frac{1}{2}[/tex]
The instructor of my course said I could use software on any of the problems in the handout but even when I tried plugging the equations into mathematica I got an error. I just don't know where to go with the equations.