What is the maximum vertical distance between the line
y = x + 56
and the parabola
y = x2 for −7 ≤ x ≤ 8?

The difference, I'll call d, vertically between the functions is:

d=x+56-x^2

Now we can take the derivative of d with respect to x:

dd/dx=-2x+1

dd/dx=0 when -2x+1=0, x=1/2

d(1/2)=x+56-x^2=56.25

Now you can check the endpoints just in case...

d(-7)=0 and d(8)=-16

So the maximum vertical distance of 56.25 units occurs when x=1/2