Monotonicity Math Example 2
Follow the full solution, then compare it with the other examples linked below.
Example 2
hardDetermine the intervals on which \(h(x) = x^3 - 3x\) is increasing and decreasing.
Solution
- 1 Find the derivative: \(h'(x) = 3x^2 - 3 = 3(x^2 - 1) = 3(x-1)(x+1)\).
- 2 Set \(h'(x) = 0\): \(x = 1\) or \(x = -1\).
- 3 Sign of \(h'(x)\): for \(x < -1\): \(h'(x) > 0\) (increasing); \(-1 < x < 1\): \(h'(x) < 0\) (decreasing); \(x > 1\): \(h'(x) > 0\) (increasing).
- 4 Increasing on \((-\infty,-1) \cup (1,\infty)\), decreasing on \((-1,1)\).
Answer
Increasing on \((-\infty,-1)\) and \((1,\infty)\); decreasing on \((-1,1)\)
A function increases where its derivative is positive and decreases where its derivative is negative. The critical points \(x = \pm 1\) divide the real line into three intervals.
About Monotonicity
A function or sequence that consistently moves in one direction only—always increasing or always decreasing throughout its domain.
Learn more about Monotonicity →More Monotonicity Examples
Example 1 medium
Is (f(x) = 2x + 3) monotonically increasing? Show that if (x_1 < x_2) then (f(x_1) < f(x_2)).
Example 3 mediumFor (f(x) = -x + 5), is it increasing or decreasing? Verify with two test values.
Example 4 hardShow that (f(x) = x^2) is NOT monotone on all of (mathbb{R}) by giving a counterexample, then state