What is the smallest n for which...

On my latest assignment in high school calculus to do over the Spring Break, I've been given this problem: "What is the smallest n for which d^n(12x^3 + 2x) / dx^n = 0, for all x?" - I don't know where to start with this, but my teacher mentioned something about finding the derivatives. Can someone here please help point me in the right direction?

Re: What is the smallest n for which...

Well, find $\displaystyle \frac{d}{dx}(12x^3 + 2x)$, $\displaystyle \frac{d^2}{dx^2}(12x^3 + 2x)$, $\displaystyle \frac{d^3}{dx^3}(12x^3 + 2x)$, ... and see when it becomes 0. Of course, you need to know what the definition of $\displaystyle \frac{d^n}{dx^n}(12x^3 + 2x)$ is.

Re: What is the smallest n for which...

Quote:

Originally Posted by

**emakarov** Well, find $\displaystyle \frac{d}{dx}(12x^3 + 2x)$, $\displaystyle \frac{d^2}{dx^2}(12x^3 + 2x)$, $\displaystyle \frac{d^3}{dx^3}(12x^3 + 2x)$, ... and see when it becomes 0. Of course, you need to know what the definition of $\displaystyle \frac{d^n}{dx^n}(12x^3 + 2x)$ is.

Now that you write it out like that, I now know exactly what the "d^n / dx^n" part meant and was able to solve the problem easily. Thanks!