I solved the equation but when it satisfies it spouse to have opposite signs i think this is the question
Question 1 The curve with equation $y=\dfrac{e^{2x}+x}{x^3}$ has a stationary point with $x$-coordinate lying between $1$ and $2$.
a. Show that the $x$-coordinate of this stationary point satisfies the equation $x=\dfrac32+\dfrac{x}{e^{2x}}$.
b. Using a suitable iterative process based on the qeuation given in part a, find the value of the $x$-coordinate of this stationary point correct to 4 decimal places. Give each iteration to 6 decimal places.
the question is 1a this is my working out. I have no teacher cso any help would be much appreciated. Thank you.
$y = \frac{e^{2x} + x}{x^3} = \frac{e^{2x}}{x^3} + \frac{x}{x^3} = e^{2x} x^{-3} + x^{-2}$. Thus you can just use the product rule:
$$\frac{dy}{dx} = 0 \Rightarrow e^{2x} \cdot -3x^{-4} + 2e^{2x} \cdot x^{-3} - 2x^{-3} = 0$$
$$\Rightarrow e^{2x} \cdot -3 + 2e^{2x} \cdot x - 2x = 0$$
then divide both sides by $e^{2x}$.