Investigating functions with the derivative
Answers are for parents and teachers.
1
Find the intervals of increase and decrease and the local extrema of f(x) = x³ − 3x² − 9x + 2.
f′ = 3(x + 1)(x − 3). Increasing on (−∞, −1) and (3, ∞); decreasing on (−1, 3). Local maximum f(−1) = 7, local minimum f(3) = −25.
2
Find the greatest and least values of f(x) = x⁴ − 2x² + 5 on [−2, 3].
f′ = 4x(x² − 1) ⇒ x = −1, 0, 1. f(−2) = 13, f(−1) = 4, f(0) = 5, f(1) = 4, f(3) = 68. Greatest 68, least 4.
3
To sketch f(x) = −x³ + 3x², find the stationary points, extrema and intersections with the axes.
f′ = −3x² + 6x = −3x(x − 2): min f(0) = 0, max f(2) = 4. Meeting Ox: x²(3 − x) = 0 ⇒ x = 0 (touching) and x = 3. Meeting Oy: (0, 0).
4
Find the minimum point of f(x) = x + 4/x (x > 0).
f′ = 1 − 4/x² = 0 ⇒ x = 2; f′ < 0 on (0, 2) and f′ > 0 on (2, ∞): minimum f(2) = 4.
5
For a function f whose derivative’s graph crosses the axis at x = −3 and x = 1, with f′ < 0 on (−3, 1) and positive elsewhere, identify the extrema.
At x = −3, f′ goes from + to −: local maximum; at x = 1 it goes from − to +: local minimum.