Let M and N be the number of points on the curve y ^ 5 - 9xy + 2x =...

Given,

${y}^{5} - 9xy + 2x = 0 . . . . . . . \left(1\right)$

Now differentiating both side we get,

$5{y}^{4}\dfrac{dy}{x} - 9x\dfrac{dy}{dx} - 9y + 2 = 0$

$⇒ \dfrac{dy}{dx}\left(5{y}^{4} - 9x\right) = 9y - 2$

$⇒ \dfrac{dy}{dx} = \dfrac{9y - 2}{5{y}^{4} - 9x} = 0 . . . . . \left(2\right)$

$⇒ 9y - 2 = 0$ (for horizontal tangent)

$⇒ y = \dfrac{2}{9}$, which does not satisfy the original equation so $M = 0$.

As ${\left(\dfrac{2}{9}\right)}^{5} - 9x \times \dfrac{2}{9} + 2x = 0$

$⇒ {\left(\dfrac{2}{9}\right)}^{5} - 2x + 2x = 0$

$⇒ {\left(\dfrac{2}{9}\right)}^{5} = 0\left\{\text{which is a false statement}\right\}$

Now $5{y}^{4} - 9x = 0$ (from equation \left(2\right)$for vertical tangent)</p><p>$⇒ 5{y}^{4}\left(9y - 2\right) - 9{y}^{5} = 0$(putting the value of$x$from equation \left(1\right)$)

$⇒ {y}^{4}\left[45y - 10 - 9y\right] = 0$

$⇒ y = 0$ or $36y = 10$$⇒ y = \dfrac{5}{18}$

So at $y = 0 ⇒ x = 0$ and $y = \dfrac{5}{18} ⇒ x = {x}_{1}$

$\begin{cases}\left(0, 0\right) & \left({x}_{1}, \dfrac{5}{18}\right)\end{cases}$ (where ${x}_{1} = \dfrac{{\left(\dfrac{5}{18}\right)}^{5}}{9 \times \dfrac{5}{18} - 2}$)

So, $N = 2$

$M + N = 0 + 2 = 2$