Question

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

Answer 1

See explanation

Explanation:

Gradient of the given equation is it differential

Thus `f^'(x)= 21x^2+18x`

`=>f^'(-2)=21(-2)^2+18(-2) = +48`

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

`y=-1/48x+c`......................................(1)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Let the point on the original curve be `P_1->(x_1,y_1)=(-2,y)`

`f(-2)=y_1=7(-2)^3+9(-2)^2=-20`

`=>P_1->(x_1,y_1)=(-2,-20)`

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

I will let you do that!