# What is the equation of the normal line of f(x)=x^3-x^2+6x-1 at x=5?

Jan 17, 2017

$x + 71 y - 9154 = 0$. See the normal-inclusive Socratic second graph.

The first is a not-to-scale graph.

#### Explanation:

graph{x^3-x^2+6x-1 [-20, 20, -150, 150]} At x = 5, y = 129.

$f ' = 3 {x}^{2} - 2 x + 6 = 71$, at $x = 5.$

Slope of the normal is #-1/f'=-1/71.

The equation to the nthe normal is

y-129=-1/71(x-5), giving

$x + 71 y - 9154 = 0$

The normal=inclusive-graph below. is for uniform scale near the foot

of the normal (5, 129), shown as a small circle.

graph{(x^3-x^2+6x-1-y)(x+71y-9154)((x-5)^2+(y-129)^2-.1)=0 [-20, 20,, 120, 140]}