Consider the parabola y=x^2-2px+p+1.

Question

If the real number p has the property a<p<b, then the vertex of y is above the line y=−12x+13.
What are a and b?

Answer 1

The vertex lies at (p,-p^2+p+1). So, we need

-p^2+p+1 > -12p+13
p^2+11p-12 < 0
-12 < p < 1

Answer 2

1<p<12
Therefore
a=1
b=12