I have following profit function -
$$ max_{x} ~ mx^{2a} - rx $$ $\text{Subject to,}$
$$ p \geq mx^{2a} - rx \geq q $$
Where, $ m>r, p>0, q>0$ and $a< \frac{1}{2}$
Since firm always want to choose higher enough profit, upper boundary constraint will be binding.
So, the maximization problem becomes - $$ max_{x} ~ mx^{2a} - rx $$ $\text{Subject to,}$ $$ mx^{2a} - rx = p $$
How can I solve for optimal value of $x$?
My working - $$ mx^{2a} - rx = 0 \Leftrightarrow x= \left(\frac{m}{r} \right)^{\frac{1}{1-2a}}$$ Since $p>0$ So, $x> \left(\frac{m}{r} \right)^{\frac{1}{1-2a}}$
My question is - is there a way to solve for exact expression of x?