I am given the following task:
The graph of the polynomial function $f(x)$ is symmetric to the y-axis. It has exactly one local minimum on the x-axis and an inflection point at $x = 1$. Find the the only possible function $f$.
My thoughts: Because $f$ is symmetric and it only has one minimum on the x-axis, this has to be at $x = 0$. Therefore: $f(0) = 0$, $f'(0) = 0$. Because of the inflection point, $f''(1) = 0$.
Now suppose $\deg(f) = 4$ (could also be $6, 8, \ldots$). Then $f(x) = ax^4+bx^2 +c$. From the above, I get $c = 0$ and $b = -6a$.
So I have multiple functions $f_a(x) = ax^4-6ax^2$, although the task stated that there is only one solution.
I can't find my mistake (or is the homework wrong?).
Thanks in advance.