Question

Find the point(s) on the curve of y=ln x at which the curvature is largest.

Answer 1

\[k(x)=\frac{ y'' }{ [1+(y')^2]^{3/2} }\]

Answer 2

\[y'=\frac{ 1 }{ x } \]

Answer 3

\[y''=-\frac{ 1 }{ x^2 }\]

Answer 4

\[k(x)=\frac{ x }{ (x^2+1)^{3/2} }\]

Answer 5

\[k'(x)=\frac{ (x^2+1)^{1/2}(1-2x^2) }{ (x^2+1)^{3} }\]

Answer 6

\[(x^2+1)^{1/2}(1-2x^2)=0\]

Answer 7

\[x^2=\frac{ 1 }{ 2 }\]

Answer 8

\[x=\pm \frac{ \sqrt{2} }{ 2 }\]

Answer 9

Those are the critical points. Now what? @SithsAndGiggles

Answer 10

\[y=\ln x\implies y'=\frac{1}{x}\implies y''=-\frac{1}{x^2}\]
so the curvature is
\[k(x)=\frac{\left|-\dfrac{1}{x^2}\right|}{\left(1+\dfrac{1}{x^2}\right)^{3/2}}\]
Some algebraic manipulation gives us the equivalent expression,
\[k(x)=\frac{1}{\left(\dfrac{1}{x^2}\right)^{1/2}\left(x^2+1\right)^{3/2}}\]
Now, \(\sqrt{x^2}=|x|\), but since \(y=\ln x\) is defined for \(x>0\), we have \(|x|=x\), so
\[k(x)=\frac{x}{\left(x^2+1\right)^{3/2}}\]
so we are in agreement with the curvature.

Answer 11

Derivative:
\[\begin{align*}k'(x)&=\frac{(x^2+1)^{3/2}-3x^2(x^2+1)^{1/2}}{(x^2+1)^3}\\\\&=\frac{(x^2+1)^{1/2}(1-2x^2)}{(x^2+1)^3}\\\\
&=\frac{1-2x^2}{(x^2+1)^{5/2}}
\end{align*}\]
Finding the critical points is a matter of solving
\[1-2x^2=0\]
which, as you've shown, has solutions \(x=\pm\dfrac{1}{\sqrt2}\), but since \(\ln x\) is defined for \(x>0\), we can ignore the negative root.

Answer 12

Now we apply the derivative test. We're checking the sign of \(k'(x)\) over two intervals, \(\left(0,\dfrac{1}{\sqrt2}\right)\) and \(\left(\dfrac{1}{\sqrt2},\infty\right)\).

If \(k'(x)>0\), this means \(k\) is increasing on the interval; if \(k'(x)<0\), then \(k\) is decreasing. If you can establish that you have a pattern of increasing followed by decreasing, this indicates that a maximum occurs at the critical point.

To check the sign on either interval, you pick a test value within the interval. Since \(\dfrac{1}{\sqrt2}\approx0.707\), you can try \(\dfrac{1}{2}\) for the first interval and \(1\) for the second. Plug these test values into \(k'(x)\).

Answer 13

You can review how to apply the test here: http://tutorial.math.lamar.edu/Classes/CalcI/AbsExtrema.aspx