Practice Monotonicity in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
A function or sequence that consistently moves in one direction only—always increasing or always decreasing throughout its domain.
Your age is monotonically increasing—it only goes up, never back down. A timer counting down is monotonically decreasing.
Example 1
mediumIs \(f(x) = 2x + 3\) monotonically increasing? Show that if \(x_1 < x_2\) then \(f(x_1) < f(x_2)\).
Example 2
hardDetermine the intervals on which \(h(x) = x^3 - 3x\) is increasing and decreasing.
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 where it IS monotone.