Question

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

Answer 1

`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.