A certain farm in Suwon produces a certain amount of bananas. This is given by the expresion from below:
$$Q(x)=12 \sec ^2 x +48\tan x +36$$
This represents hundreds of US dollars.
Assuming $x$ is given by an acute angle, and we know that the utility of such farm is an integer number of US dollars. Find the least utility of such farm.
The alternatives given in my book are as follows:
$\begin{array}{ll} 1.&\textrm{4801 usd}\\ 2.&\textrm{4851 usd}\\ 3.&\textrm{4849 usd}\\ 4.&\textrm{4999 usd}\\ \end{array}$
What I did in my attempt to find the least was to complete the square and use this fact:
$(a)^2\geq 0$
And: $1+\tan^2 \omega = \sec^2 \omega$
Then replacing these in the aforementioned utility function I'm getting:
$Q(x)=12 \sec ^2 x +48\tan x +36$
$Q(x)=12(1+\tan^2 x)+48\tan x +36$
$Q(x)=12+12\tan ^2x +48\tan x +36$
$Q(x)=\left(\sqrt{12}\tan x +\frac{24}{\sqrt{12}}\right)^2$
Then this becomes into:
$\left(\sqrt{12}\tan x +\frac{24}{\sqrt{12}}\right)^2 \geq 0$
Then it mentions that $x$ is an acute angle therefore:
$0<x<\frac{\pi}{2}$
Then I assumed that:
$\tan 0 < \tan x <\tan\frac{\pi}{2}$
$0< \tan x < \infty$
Then I completed the interval as it was given by the mentioned square:
$0 < \sqrt{12}\tan x < \infty$
$\frac{24}{\sqrt{12}} < \sqrt{12}\tan x + \frac{24}{\sqrt{12}} < \infty$
$48 < \left(\sqrt{12}\tan x + \frac{24}{\sqrt{12}}\right)^2 < \infty$
Since the function represents hundreds of US dollars then I assumed I had to multiply by $100$ the earlier inequation to obtain the least profit or utility.
$4800 < \left(\sqrt{12}\tan x + \frac{24}{\sqrt{12}}\right)^2 < \infty$
Since it mentions that the least profit or utility is integer then I assumed that this would be $4801$ which is the next number or the local minima. But is this a right assumption?
This result appears in one of the alternatives. But I'm not sure if my rationale is correct.
The reason of this is that I had to multiply for $100$ to one side of the inequality, not the other terms, in other words the part which is in the center and the infinity. For obvious reasons it doesn't really matter much to multiply to infinity because it will remain the same, but for the part which is the function in the center, this confuses me.
So all and all is my method correct?. Can someone explain this me better?. The thing is I'm looking for a method which would refrain from using derivatives and approach this using precalculus tools.
Therefore, can someone help me?. Please I require step by step explanation because I often get a little bit tangled with equations and concepts.