# What are the inflection points of f(x)=9\sin x +\cot x?

## Question:

What are the inflection points of {eq}f(x)=9\sin x +\cot x{/eq}?

## Inflection point

Trigonometric functions on the cartesian plane have inflection point at the second derivative is zero, remember, the trigonometric functions have repetitive behavior.

The function is:

{eq}\ f(x) = 9\sin x +\cot x \\ {/eq}

But, {eq}\displaystyle \cot x= \frac{cos(x)}{sin(x)} \\ {/eq}

So, the function is:

{eq}\displaystyle \ f(x) = 9\,\sin \left( x \right) +{\frac {\cos \left( x \right) }{\sin \left( x \right) }} \\ {/eq}

Its first derivative is:

{eq}\displaystyle \ f'(x)= 9\,\cos \left( x \right) -1-{\frac { \left( \cos \left( x \right) \right) ^{2}}{ \left( \sin \left( x \right) \right) ^{2}}} \\ \displaystyle \ f'(x)= -{\frac {9\, \left( \cos \left( x \right) \right) ^{3}-9\,\cos \left( x \right) +1}{ \left( \sin \left( x \right) \right) ^{2}}} \\ {/eq}

Its second derivative is:

{eq}\displaystyle \ f''(x) = -9\,\sin \left( x \right) +2\,{\frac {\cos \left( x \right) }{\sin \left( x \right) }}+2\,{\frac { \left( \cos \left( x \right) \right) ^{3}}{ \left( \sin \left( x \right) \right) ^{3}}} \\ \displaystyle \ f''(x) = -{\frac {9\, \left( \cos \left( x \right) \right) ^{4}-18\, \left( \cos \left( x \right) \right) ^{2}-2\,\cos \left( x \right) +9}{ \left( \sin \left( x \right) \right) ^{3}}} \\ {/eq}

Inflection point at {eq}\displaystyle \ f''(x) =0 \; \Rightarrow \; 0= -{\frac {9\, \left( \cos \left( x \right) \right) ^{4}-18\, \left( \cos \left( x \right) \right) ^{2}-2\,\cos \left( x \right) +9}{ \left( \sin \left( x \right) \right) ^{3}}} \\ \displaystyle 0= -(9\, \left( \cos \left( x \right) \right) ^{4}-18\, \left( \cos \left( x \right) \right) ^{2}-2\,\cos \left( x \right) +9 ) \; \Leftrightarrow \; x \; \approx \; 0.6976272891409 \\ {/eq}

The function has inflection point at:

{eq}\displaystyle (0.6976272891409, f(0.6976272891409)) \; \Rightarrow \; (0.6976272891409, 6.974585279) \\ {/eq}

Therefore, the inflection point are:

{eq}\displaystyle x= 2\pi \mathbb{N}+0.6976272891409 \; \; \; \text{or} \\ \displaystyle (2\pi \mathbb{N}+0.6976272891409, f(2\pi \mathbb{N}+0.6976272891409)) \\ {/eq}