How to solve linnear approximation - a calculus problem

Question

Let $f(x) = 2x^3+(0)x^2+ (-6)x.$ Find $x=c$ so that the linear approximation $g(x)$ to $f(x)$ in $x = c$ for $g(0) = -108$

I really can't solve this problem and it has been bugging me for hours. Can someone please explain how to solve this?

Answer 1

If $f(x)=2x^3-6x$, its linear approximation around $x=c$ is given by $$ g(x)= f(c) + f'(c)(x-c) = 2c^3-6c + (6c^2-6) (x-c). $$

Se $g(0)=-108$, you have that $$ 2c^3-6c-6c^3+6c = -108 \Leftrightarrow c^3 = 27 \Leftrightarrow c = 3. $$