Finding Critical Numbers

Question

Find the critical numbers of the given function: f (x) = 2 sec (x) + tan (x) ; 0 < x < 2pi

Answer 1

Note that the numbers π/2 and 3π/2 are not in the domain of the function.

f'(x) = 2sec(x)tan(x) + sec²(x)

Find the critical numbers by setting f' equal to zero and solving for x. Critical numbers also occur where the first derivative is undefined, but are in the domain of the function.

sec(x)[2tan(x) + sec(x)] = 0

sec(x) = 0 (never happens)

or

2tan(x) + sec(x) = 0

2sin(x)/cos(x) + 1/cos(x) = 0 (we know cos(x) ≠ 0, because π/2 and 3π/2 are not in the function's domain)

2 sin(x) + 1 = 0 (multiply every term by cos(x))

sin(x) = -1/2

x = arcsin(-1/2)

x = 7π/6 + 2kπ, 11π/6 + 2kπ, k is any integer. When k = 0, x is restricted to 0<x<2π.

So the critical numbers are 7π/6 and 11π/6.

The graph of the original function has horizontal tangent lines at the points with those x values:

desmos.com/calculator/iovu6kptls