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Chapter 11: Problem 68

Find the curvature of \(f(x)=\ln x,\) for \(x>0\) and find the point at which it is a maximum. What is the value of the maximum curvature?

Short Answer

Expert verified
Answer: To find the maximum curvature of the function \(f(x) = \ln x\), follow the steps outlined in the solution. First, find the first and second derivatives of the function, and then use the formula for curvature to find the curvature function \(\kappa(x)\). Next, find the critical points by setting the derivative of the curvature equal to zero and solve for x. Apply the second derivative test to determine if there is a local maximum at those points. Finally, find the maximum curvature and the point at which it occurs by plugging the x-coordinate into the curvature function and the original function.

Step by step solution

01

Calculate the first derivative of f(x)

To find the first derivative, use the basic rules of differentiation. Since \(f(x) = \ln x\), we can use the derivative of ln(x) which is \(\frac{1}{x}\). Therefore, the first derivative of the function is \(f'(x) = \frac{1}{x}\).
02

Calculate the second derivative of f(x)

Now, find the second derivative of the function \(f(x)\) by differentiating \(f'(x) = \frac{1}{x}\) with respect to x. The derivative of \(\frac{1}{x}\) is \(-\frac{1}{x^2}\). Thus, the second derivative of the function is \(f''(x) = -\frac{1}{x^2}\).
03

Find the curvature of f(x)

The curvature of a function is given by the formula \(\kappa(x) = \frac{|f''(x)|}{(1 + (f'(x))^2)^{\frac{3}{2}}}\). Using the values of \(f'(x)\) and \(f''(x)\) from steps 1 and 2, the curvature of f(x) is: \(\kappa(x) = \frac{|\frac{-1}{x^2}|}{\left(1 + \left(\frac{1}{x}\right)^2\right)^{\frac{3}{2}}}=\frac{\frac{1}{x^2}}{\left(1 + \frac{1}{x^2}\right)^{\frac{3}{2}}}\)
04

Find the maximum curvature

To find the maximum value of the curvature, we need to analyze the function \(\kappa(x)\) for local extrema. By differentiating \(\kappa(x)\) with respect to x, and setting the derivative equal to zero, we can find the x-coordinate(s) at which the extrema occur. The next step is to use the second derivative test to determine if the extrema are indeed maxima.
05

Simplify the expression for the curvature and find its derivative

Before differentiating, simplify the expression for the curvature: \(\kappa(x) = \frac{\frac{1}{x^2}}{\left(1 + \frac{1}{x^2}\right)^{\frac{3}{2}}}=\frac{1}{x^2\left(1 + \frac{1}{x^2}\right)^{\frac{3}{2}}}\) Now, find the derivative of the curvature with respect to x, denote this as \(\kappa'(x)\).
06

Set the derivative of the curvature to zero and solve for x

Set the derivative of curvature, \(\kappa'(x)\), equal to zero and solve for x. This will give you the critical points of the curvature function.
07

Apply the second derivative test

Find the second derivative of the curvature function, \(\kappa''(x)\). If the second derivative at the critical points found in Step 6 is negative, then there is a local maximum at that point.
08

Find the maximum curvature and the point at which it occurs

Plug the x-coordinate of the local maximum into the curvature function \(\kappa(x)\) to find the maximum curvature. The point at which the maximum curvature occurs is (x, f(x)), where x is the x-coordinate of the local maximum and f(x) is the value of the function at that x-coordinate.

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