Problem 3. Students were asked the following: find all points where the graph of the function
f (ac) = (202 - 1) (23 - 9202 + 242 + 92)
has a horizontal tangent line. Explain why the three solutions below are wrong, then give the
correct strategy. Note: you'll likely need a calculator to solve this problem completely, so the
strategy is enough.
(a) Horizontal tangent line means that we need to solve f (x) = 0. This means that x2 -1 = 0
or x3 - 9x2 + 24x + 92 = 0. Solving for x, we get x = -1, 1 for the first equation and
x = -2 for the second equation (which only has one root).
(b) Horizontal tangent line means that we need to find f'(0). So first we take the derivative,
f' (20) = (202 - 1) (203 - 9202 + 24x + 92) = (20) (3x2 - 18x + 24) = 6x3 - 36x + 48x
We therefore conclude that f' (0) = 0.
(c) Horizontal tangent line means that we need to solve f'(x) = 0. So first we take the
derivative,
f' (ac) = - (202 - 1) (23 - 9202 + 242 + 92) = (2x) (3x2 - 18x + 24) = 6x3 - 36x+48x
and then factor it, so we need to solve 6x(x2 - 6x + 8) = 0. We can solve this to find the
solutions are x = 0, 2, 4.