Theory question 112
Differentiation(a) Find the maximum and minimum points of the curve y = 2x3 − 3x2 − 12x + 4.
(b) Sketch the curve in 12(a) above.
Show the model answer
Model answer
(a) dy/dx = 6x² − 6x − 12 = 6(x − 2)(x + 1) = 0, so x = 2 or −1.
d²y/dx² = 12x − 6: at x = −1 it is −18 (maximum), at x = 2 it is 18 (minimum).
Maximum point (−1, 11); minimum point (2, −16).
(b) Cubic rising from the left to (−1, 11), down through (0, 4) to (2, −16), then rising again.
Explanation
Use the second derivative to tell the maximum from the minimum.