To evaluate the solution, I integrated the equation $f'''(x)=0$ thrice, which yields $f(x)$ as a quadratic polynomial (with zero as a permissible coefficient for all the terms).
My questions are:
Is this the only type of function whose third derivative vanishes?
I was wondering if there was a possibility of other functions (trigonometric, logarithmic, exponential, or even non-elementary) that exhibit this property.
If the only permissible solution is a quadratic or lower degree polynomial, I would like a proof which rules out the possibility of any other function to exhibit this behaviour.
In essence it is the fundamtal theorem of calculus. I marked the points where the theorem is used with a exclamation mark. $$ f'''(x)=0\\ \Rightarrow \int_a^x f'''(x) dx \stackrel{!}{=} f''(x)-f''(a)= 0\\ \Rightarrow \int_a^x f''(x)-f''(a) dx \stackrel{!}{=} f'(x)-f'(a)-f''(a)(x-a)= 0\\ \Rightarrow \int_a^x f'(x)-f'(a)-f''(a)(x-a) dx \stackrel{!}{=} f(x)-f(a)-f'(a)(x-a)-\tfrac{1}{2}f''(a)(x-a)^2= 0\\ \Leftrightarrow f(x)=f(a)+f'(a)(x-a)+\tfrac{1}{2}f''(a)(x-a)^2 $$