The problem is as follows:

> 
> In a pharmaceutical lab in Daegu, a group of scientists need to find
> the power of a visible blue light lamp to get through a sucrose
> crystal in order to measure its purity for a certain medication.
> 
> 
> The power for such light is given by:
> 
> 
> $$J=\frac{1}{H}$$
> 
> 
> Assuming that $H$ has to be the minimum value of the equation from
> below:
> 
> 
> $$H=9\tan\phi+4\cot\phi+3$$
> 
> 
> $\theta$ is an acute angle which lies in the first quadrant.
> 
> 
> What would be the value of $J$ so that the scientists could set up
> their lamp?
> 
> 
> 

The alternatives given in my book are:

$\begin{array}{ll}
1.&\frac{1}{5}\\
2.&\textrm{1}\\
3.&\textrm{15}\\
4.&\frac{1}{15}\\
\end{array}$

This problem is making me feel confused. Should derivatives be used here?

I've attempted to do so and derivate the equation, by doing so this yields:

$H'=9\sec^2\phi-4\csc^2\phi$

But equating this to zero doesn't seem to be the intended approach.

Then I came up with the idea of adding the intervals from where $\phi$ in the first quadrant lies.

$0\leq \cot \phi < +\infty$

Tangent is also in the same interval:

$0\leq \tan \phi < +\infty$

Then multiplying by $9$ and by $4$ and adding $3$ to the second equation.

This results into:

$3\leq 9\tan\phi+4\cot\phi+3 < +\infty$

**But without any further constrain there isn't any way to tell, which would be the least value?**

If it would had been the integer value, wouldn't had been $4$?

Running out of idead, what it came to my mind was to equate both functions:

$\tan\phi=\cot\phi$

This corresponds to $\phi=\frac{\pi}{4}$

Assuming this: *I'm guessing that the result would be*

$9\tan\phi=1$

$4\cot\phi=1$

then $1+1+3=5$

But this is more like guessing and not a serious way to justify the answer. I cannot be sure if that's the answer. Can someone help me here and **explain me why derivatives doesn't seem to work?**

**How to solve this without derivative?**. Help me please, I'm confused.