Question

We can repeat the derivative operation: the second derivative $f''$ is the derivative of $f'$, the third derivative $f'''$ is the derivative of $f''$, and so on. Find a function $f$ such that $f'$ and $f''$ are not the function 0, but $f'''$ is the constant function 0.

Answer 1

f(x) = x^2

Explanation:

Any quadratic (2nd-degree) function will have 0 as its third derivative.

f(x) = ax² +bx +c

f'(x) = 2ax +b

f''(x) = 2a

f'''(x) = 0

Perhaps the simplest such function is f(x) = x^2.