# What is the equation of the line normal to f(x)=7x ^3+9x^2  at x=-2?

Jul 1, 2016

See explanation

#### Explanation:

Gradient of the given equation is it differential

Thus ${f}^{'} \left(x\right) = 21 {x}^{2} + 18 x$

$\implies {f}^{'} \left(- 2\right) = 21 {\left(- 2\right)}^{2} + 18 \left(- 2\right) = + 48$

So the gradient of the line normal to it at this point is:

$y = - \frac{1}{48} x + c$......................................(1)
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Let the point on the original curve be ${P}_{1} \to \left({x}_{1} , {y}_{1}\right) = \left(- 2 , y\right)$

$f \left(- 2\right) = {y}_{1} = 7 {\left(- 2\right)}^{3} + 9 {\left(- 2\right)}^{2} = - 20$

$\implies {P}_{1} \to \left({x}_{1} , {y}_{1}\right) = \left(- 2 , - 20\right)$

Substitute these values into equation (1) to find $c$

I will let you do that!