Critical numbers of trig functions

[jeonw]

1. The problem statement, all variables and given/known data
Find the critical Numbers

f(theta)=2sec(theta)+tan(theta)

2. The attempt at a solution
I found the derivative and set it equal to zero and to reduce writing I substituted x for theta
f'(x)=2sec(x)tan(x)+sec^2(x)
factored out the sec(x)
sec(x)[2tan(x)+sec(x)]=0

My question is where are the critical numbers? do the critical numbers exist where sec(x) is underfined because sec(x) will never equal 0.

 

[soroban]

Hello, jeonw!

You're doing fine . . .

1. Find the critical numbers of: .\(\displaystyle f(x) \:=\:2\sec x + \tan x\)


The attempt at a solution:

I found the derivative and set it equal to zero: .\(\displaystyle f'(x)\:=\:2\sec x\tan x+\sec^2x \:=\:0\)

Factored: .\(\displaystyle \sec x (2\tan x +\sec x ) \:=\:0\)

Set each factor equal to zero and solve.


\(\displaystyle \sec x \:=\:0\;\hdots \text{ impossible.}\)


\(\displaystyle 2\tan x + \sec x \:=\:0 \quad\Rightarrow\quad \frac{2\sin x}{\cos x} + \frac{1}{\cos x} \:=\:0 \quad\Rightarrow\quad \frac{2\sin x +1}{\cos x}\:=\:0\)

. . \(\displaystyle 2\sin x + 1 \:=\:0 \quad\Rightarrow\quad \sin x \:=\:-\tfrac{1}{2}\)


\(\displaystyle \text{Therefore: }\;x \;=\;\begin{Bmatrix}\text{-}\dfrac{\pi}{6} + 2\pi n \\ \\[-3mm] \dfrac{7\pi}{6} + 2\pi n \end{Bmatrix}\;\text{ for any integer }n.\)