I have the following function:
$$ f(x) = {\sin{x}\over a + \cos{x}} $$
I have found that this function has a maximum $$ f_\mathrm{max} = \arctan{\left(1\over\sqrt{a^2 - 1}\right)}$$ at $x = \arccos{\left(-{1\over a}\right)}$.
However, I am considering now the function:
$$ g(x) = |f(x) + b| $$
And I am trying to find the maximum value of $g(x)$ and its corresponding $x$ value.
Using the graph of the function, I can see that:
$$ g_\mathrm{max} = f_\mathrm{max} + |b| $$
But this time, the $x$ coordinate of the $g_\mathrm{max}$ will depend on the sign of $b$. That is:
- if $b \geq 0$ then $g_\mathrm{max}$ occurs at: $x = \arccos{\left(-{1\over a}\right)}$
- if $b < 0$ then $g_\mathrm{max}$ occurs at $x = 2\pi - \arccos{\left(-{1\over a}\right)}$
My question:
Is it possible to express the last two conditions into a single formula? in other words, is there a single formula for $x$ coordinate of $g_\mathrm{max}$ ?
Your help is very much appreciated