Question

The curve with equation y=e^2x/4+3e^x has one stationary point.find the exact value of the coordinates of this point

Answer 1

$y=\dfrac{e^{2x}}{4+3e^x}$
$\implies y'=\dfrac{2e^{2x}(4+3e^x)-3e^xe^{2x}}{(4+3e^x)^2}$
$\implies y'=\dfrac{8e^{2x}+3e^{3x}}{(4+3e^x)^2}$

Stationary points occur where the derivative is zero. The denominator is positive for all $x$, so we only need to worry about the numerator.

$8e^{2x}+3e^{3x}=e^{2x}(8+3e^x)=0$

$e^{2x}>0$ for all $x$, so we can divide through:

$8+3e^x=0\implies e^x=-\dfrac83$

But $e^x>0$ for all $x\in\mathbb R$, so this function has no stationary points...

I suspect there may be a typo in the question.