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

May 16, 2016

$f \left(x\right) = 5 {x}^{3} + {x}^{2} - 5 x - 2$

At $x = 2$, we have $y = f \left(2\right) = 44 - 12 = 32$

$f ' \left(x\right) = 15 {x}^{2} + 2 x - 5$, so the slope of the tangent line is $f ' \left(2\right) = 64 - 5 = 59$

The normal line is perpendicular to the tangent, so its slope is $- \frac{1}{59}$

An equation for the normal line is $y = 32 - \frac{1}{59} \left(x - 2\right)$.

This can also be written $y = - \frac{1}{59} x + 32 \frac{2}{59}$.