The number of points on the curve [y=54 x^5-135 x^4-70 x^3+180 x^2+210 x] at which the nor…

Question

The number of points on the curve $$y=54 x^{5}-135 x^{4}-70 x^{3}+180 x^{2}+210 x$$ at which the normal lines are parallel to $x+90 y+2=0$ is :

Accepted Answer

$y'=270x^4-540x^3-210x^2+360x+210$

Slope of normal $$=-\frac{1}{90}$$

$$\therefore$$ Slope of tangent = 90

$$\therefore$$ Number of normal will be number of solutions of

$270x^4-540x^3-210x^2+360x+210=90$

$$\Rightarrow 9x^4-18x^3-7x^2+12x+4=0$$

$$\therefore x=1,2,-\frac{1}{3},-\frac{2}{3}$$ are roots

Solutions · 1