Calculus of a Single Variable, Chapter 5, 5.8, Section 5.8, Problem 40

Relative extrema of a function are located at the points where the derivative of the function is zero or does not exist.
The derivative of h(x) = 2tanx - x is
h'(x) = 2sec^2x - 1 . The h'(x) does not exist at the points where sec(x) is undefined, but at these points tan(x) is also undefined, so there cannot be extrema there. The extrema will occur at the points where h'(x) = 0.
2sec^2(x) - 1 = 0
sec^2(x) = 1/2
sec(x) = +-sqrt(2)/2 = +-0.7
By definition of secant, sec(x) has to always be greater than 1 or less than -1, so the equation
sec(x) = +-sqrt(2)/2
has no solutions. The given function h(x) = 2tan(x) - x has no relative extrema. Please see the attached image for the illustration.