# What is the equation of the line normal to f(x)=2x^3 - x^2-x  at x=4?

May 21, 2018

$R \left(x\right) = \frac{4 - x}{87} + 108$

#### Explanation:

$f \left(4\right) = 2 \cdot 64 - 16 - 4 = 128 - 20 = 108$

$f ' \left(x\right) = 6 {x}^{2} - 2 x - 1$

$f ' \left(4\right) = 6 \cdot 16 - 8 - 1 = 96 - 9 = 87$

$R \left(x\right) = - \frac{1}{87} x + b$ is the desired perp.

$R \left(4\right) = 108 \Leftrightarrow - \frac{4}{87} + b = 108$

$b = 108 + \frac{4}{87}$