Find all the values of the parameter a, for each of which the graph of the function is symmetric with respect to the line

Question

Find all the values of the parameter $a$, for each of which the graph of the function $f(x) = x^4 -6x^3 + 12x^2 + ax$ is symmetric with respect to the line $x = a$

Answer 1

hint

The graph has the vertical line $ x=a$ is a symetry axis if only if for each one of its points $ (x,f(x)), $ its symetric

$ (2a-x,f(x) ) $ is also in the graph.

So, the condition is

$$(\forall x \in\Bbb R)\;\; f(2a-x)=f(x)$$

In particular, taking $x=0,$ we find

$$f(2a)=a^2(16a^2-48a+50)=f(0)=0$$