Calculus word problem

Question

What is the maximum vertical distance between the line y = x + 6

 and the parabola y = x² for −2 ≤ x ≤ 3?

Answer 1

The vertical distance is the difference between the y values, so we can write an equation--call the difference D:

D = x² - x - 6

To maximize or minimize D, we find the 1st derivative and set = to 0:

D' = 2x - 1 = 0

Thus x = 1/2 But--how do we know if this is a maximum or minimum? We no need to evaluate D at the 2 given points:

x = 3; D = 9 - 3 - 6 = 0

x = -2; D = 4 + 2 - 6 = 0

Thus, x = 1/2 does in fact give the maximum D, at D = (1/2)² - 1/2 - 6 = 1/4 - 2/4 -24/4 = -25/4

CHECK by plotting the functions using a graphing calculator or online tool