Question

What is the minimum value of \(|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|\)

Answer 1

Start by finding the derivative of the function. Equate the derivative to zero and then obtain a value for x. Plug the value of x in the given equation and you will have the minimum value

Answer 2

\[\rm \frac{d}{dx} |f(x)| = \frac{f(x)}{|f(x)|} \times \frac{d}{dx} f(x)\]
\[\rm \frac{d}{dx} \sin x = \cos x\]\[\rm \frac{d}{dx} \cos x = -\sin x\]\[\rm \frac{d}{dx} \tan x = \sec^2 x\]\[\rm \frac{d}{dx} \sec x = \sec x \tan x\]\[\rm \frac{d}{dx} \cot x = \csc^2 x\]\[\rm \frac{d}{dx} \csc x = -\cot x \csc x \]

Answer 3

Let see if you manage to find the stationary point(s)

Answer 4

The minimum is about 1.82843

Answer 5

Notice that the function
\[
\sin x+\cos x+\tan x+\cot x+\sec x+\csc x
\]
is positive when\[
0\le x\le \frac \pi 2
\]

Answer 6

the inequalities are strict above sorry

Answer 7

Also the above sum is negative when
\[ \frac \pi2< x < 2 \pi
\]

Answer 8

Knowing that you can split the interval into 2 intervals and get rid of the absolute values

Answer 9

The minimum in \( ] 0, 2\pi[\) occurs at
\[
x=2 \pi +\tan ^{-1}\left(\frac{1-\sqrt{2}-\sqrt{2 \sqrt{2}-1}}{1-\sqrt{2}+\sqrt{2
\sqrt{2}-1}}\right)
\]

Answer 10

OK, I got it