Here's the link to the problem:
https://artofproblemsolving.com/wiki/index.php/2010_AMC_12A_Problems/Problem_21
The graph of y=x^6-10x^5+29x^4-4x^3+ax^2 lies above the line bx-c=y except at three values of x , where the graph and the line intersect. What is the largest of these values?
My logic: The line bx-c=y must be tangent to 3 local minimums of the function. However, one of those local minimums must be 0 because x^6-10x^5+29x^4-4x^3+ax^2=x^2(x^4-10x^3+29x^2-4x+a). Therefore, the y intercept of the line must be 0. There is no other way that the line can intersect (0,0). Also, that means that 0x+c=0, c=0. So why does the official solutions not factor this in? They keep treating it even though it must be to be tangent to the local minima.
I would appreciate insight about where my logic is flawed