How do you find the critical numbers of #g(θ) = 4 θ - tan(θ)#?

1 Answer
Mar 23, 2017

Answer:

#theta=0# is an inflection point.

Explanation:

Extrema
#g'(theta) = 4 - \frac{1}{\cos^2(theta)} = 0#
or
#\cos^2(theta) = \frac{1}{4}#
#\cos(theta) = pm \frac{1}{2}#

That is
#theta = \frac{pi}{3} #
#theta = \frac{5\pi}{3}#
for #cos(theta) = \frac{1}{2}#
and
#theta = \frac{2\pi}{3}#
#theta = \frac{4\pi}{3}#
for #cos(theta) = -\frac{1}{2}#

#g''(theta) = -(-2)cos^(-3)(theta)(-sin(theta)) = -2cos^(-3)(theta)sin(theta)#

graph{4x-tan(x) [-7, 7, -7, 7]}
graph{sin(x)/cos(x)^3 [-7, 7, -30, 30]}

As you can see on the bottom plot, and as expected, the second derivative of g changes sign at x = 0. Therefore x=0 is an inflection point. The extrema are true extrema. The second derivative does not change sign at the extrema. If it did, these would not be extrema but saddle points.