Question

Find all relative extrema. Use the Second Derivative Test where applicable. \(f(x)=\operatorname{arcsec} x-x\)

Short answer

The critical points are \(x = -1, 1\). The result of the Second Derivative Test will determine whether these points represent relative maxima or minima.

Step-by-step solution

Differentiate the Function

The first step is to differentiate the function \(f(x)= arcsec x - x\). The derivative of \(arcsec x\) is \(1/(|x|\sqrt{x^2 -1})\) and the derivative of \(x\) is 1. Thus, the first derivative \(f'(x)= \frac{1}{|x|\sqrt{x^2 -1}} -1\)

Find the Critical Points

Critical points occur where the first derivative equals zero or does not exist. Solve for \(x\) in the equation \(f'(x) = 0\), by setting \(\frac{1}{|x|\sqrt{x^2 -1}} -1 = 0\). Solving this gives \(x = -1, 1\). Next, determine where \(f'(x)\) does not exist. Since the denominator can't be zero, we find \(x\) in the equation \(|x|\sqrt{x^2-1} = 0\), which has no real solutions. Therefore, the critical points are \(x = -1, 1\).

Apply Second Derivative Test

Take the second derivative of \(f(x)\), namely \(f''(x)\). Then use Second Derivative Test, i.e., plug each of critical points into the second derivative. If the second derivative at a point is positive, then \(f(x)\) has a relative minimum there. If it's negative, \(f(x)\) has a relative maximum there. If it's zero, the test is inconclusive.