Recall that a critical value of a function f(x) occurs where the derivative of f(x), denoted as either f'(x) or (df)/dx, is equal to 0 or is undefined. Thus, we must first find the derivative. First we should determine whether the function is continuous throughout its domain.
Recall that:
cot x = (cos x)/sin x and tan x = (sin x)/cos x.
Since one cannot divide by zero, the original function is discontinuous in the neighborhood of those values for which sin (x) or cos (x) = 0. This consists of any value which can be defined as npi/2, where n is an integer. Within our domain we have five such values: 0, pi/2, 2pi/2 = pi, 3pi/2, 4pi/2 = 2pi. Thus, no matter what the derivative tells us, these cannot be critical points.
Remembering that the function is undefined (and thus non-differentiable) at those five values, we shall find the general derivative of the function. Recall that the derivative of sin x is cos x. The derivatives of tan(x) and cot(x) will be quickly found below using the quotient rule.
d/dx tan(x) = d/dx (sin x/cosx) = (cos^2(x)+sin^2(x))/cos^2x = 1/cos^2x = sec^2x
d/dx cot (x) = d/dx (cos x/sin x) = (-sin^2x - cos^2x)/sin^2x = -1(sin^2x + cos^2x)/sin^2x = -1/sin^2x = -csc^2x
Thus, differentiating our original f(x) yields d/dx (cot x - sin x + tan x) = -csc^2x - cos x + sec ^2x
Our possible critical points occur where f(x) is defined, but f'(x) either equals 0 or is undefined. First we check for undefined points: in this situation these will occur whenever we must divide by 0. Since csc x = 1/sinx and sec x = 1/cos x, these occur when either sin (x) or cos (x) is equal to 0. These occur at the same points we discovered above, the iterations of npi/2. We know these cannot be critical points, so there are no critical points occurring where the derivative is undefined.
Next we will check for where f'(x) = 0. At this point it may be beneficial to utilize a graphing calculator; none of the calculations I have run without a calculator have helped to find the roots. Provided below is the appropriate graph of the derivative; the zeroes of this function between [0,2pi] are the desired critical points.
graph{-(csc(x))^2 - cos(x) + (sec(x))^2 [-0.1, 6.3, -10, 10]}
