Question

Given a twice-differentiable function \(y=f(x)\), determine its curvature at a relative extremum. Can the curvature ever be greater than it is at a relative extremum? Why or why not?

Short answer

The curvature of a twice-differentiable function \(y = f(x)\) at its relative extrema can be calculated using the curvature formula. In general, given non-zero second derivatives at the extrema, the curvature at that extrema cannot be greater elsewhere, as the curvature represents the maximum or minimum possible based on the function's concavity.

Step-by-step solution

Determine Function Derivatives

The first and second derivatives of \(y=f(x)\) will be calculated. The first derivative provides the slope of the function or tangent at a particular point, and the second derivative reveals its local maximum and minimum points, which are characterized as the relative extrema.

Solve for Extrema Points

Use the results from step 1 to set the first derivative of the function equal to zero and solve for the values of \(x\). These points, where the derivative is zero, mark the relative extrema.

Calculate the Curvature at Extrema

At each of the extrema points found in step 2, calculate the curvature which can be given by the formula, \(k(x) = \frac{\left| f''(x) \right|}{\left(1 + \left(f'(x)\right)^2\right)^{3/2}} \) where \(f'(x)\) is the first derivative, and \(f''(x)\) is the second derivative of the function. Plug in the values of \(x\) obtained in step 2 into this formula to obtain the curvatures at the extrema.

Determine if Curvature is Greater Elsewhere

Examine the graph of the function or use the formula for the curvature to determine whether it can be greater elsewhere. In general, if a function \(y=f(x)\) is twice-differentiable and the second derivative at the extrema is not zero, the curvature at that extrema is the maximum or minimum possible, depending on the sign of the second derivative at the extrema. If the function is `concave up', it will have a minimum curvature at the extrema, while if it is `concave down', it will have a maximum curvature.