Finding maximum curvature on lnx

[tensor0910]

Homework Statement

Determine the point on a plane curve f(x) = ln x where the curvature is maximum.[/B]

Homework Equations

k(x) = || T ' (x) || / || r ' (x ) ||

k (x) = f '' (x) / [ 1 + ( f '' (x))² ] ^3/2[/B]

The Attempt at a Solution

f ' (x) = 1/x
f " (x) = -1/x²

k(x) = 1/x² / { [1 + (1/x)² ] ^3/2 }

k(x) = 1/x² / { [ x² +1 /x² ] ^3/2 }

iirc we can use the first derivative and try to find the local max...but how do we even start it with this mess? quotient rule? Maybe there's something about the problem that tells us its impossible...? I'm lost here...

 

[Mark44]

Homework Statement

Determine the point on a plane curve f(x) = ln x where the curvature is maximum.[/B]

Homework Equations

k(x) = ιι T ' (x) ιι / ιι r ' (x ) [/B]

There's a lot of extra stuff that I can't read in your formula above. What are these characters? ιι

k (x) = ι f '' (x) ι / [ 1 + ( f '' (x))² ] ^3/2

And what's this one? ι

The Attempt at a Solution

f ' (x) = 1/x
f " (x) = 1/x²

You lost a sign here (above).

k(x) = 1/x² / { [1 + (1/x)² ] ^3/2 }

k(x) = 1/x² / { [ x² +1 /x² ] ^3/2 }

iirc we can use the first derivative and try to find the local max...but how do we even start it with this mess? quotient rule? Maybe there's something about the problem that tells us its impossible...? I'm lost here...

Sure, quotient rule would work, provided you are careful and methodical. It would be helpful to simplify the expression as much as possible for attempting to differentiate it.

 

[tensor0910]

I cleaned it up a bit. Sorry!

 

[ehild]

Homework Statement

Determine the point on a plane curve f(x) = ln x where the curvature is maximum.[/B]

Homework Equations

k(x) = || T ' (x) || / || r ' (x ) ||

k (x) = f '' (x) / [ 1 + ( f '' (x))² ] ^3/2[/B]

The Attempt at a Solution

f ' (x) = 1/x
f " (x) = -1/x²

k(x) = 1/x² / { [1 + (1/x)² ] ^3/2 }

k(x) = 1/x² / { [ x² +1 /x² ] ^3/2 }

iirc we can use the first derivative and try to find the local max...but how do we even start it with this mess? quotient rule? Maybe there's something about the problem that tells us its impossible...? I'm lost here...

There are some mistakes in your formulas.