Monotonicity Math Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

Show that \(f(x) = x^2\) is NOT monotone on all of \(\mathbb{R}\) by giving a counterexample, then state where it IS monotone.

1
Counterexample: \(-3 < 1\) but \(f(-3)=9 > f(1)=1\). So \(x\) increased but \(f(x)\) decreased.
2
Also: \(-3 < -1\) and \(f(-3)=9 > f(-1)=1\). Decreasing on negative side.
3
But: \(0 < 2\) and \(f(0)=0 < f(2)=4\). Increasing on positive side.
4
Monotone decreasing on \((-\infty, 0]\), monotone increasing on \([0, \infty)\).

Answer

Not monotone on \(\mathbb{R}\); decreasing on \((-\infty,0]\), increasing on \([0,\infty)\)

\(f(x) = x^2\) decreases for negative \(x\) and increases for positive \(x\), with a minimum at \(x=0\).

About Monotonicity

A function or sequence that consistently moves in one direction only—always increasing or always decreasing throughout its domain.

Is (f(x) = 2x + 3) monotonically increasing? Show that if (x_1 < x_2) then (f(x_1) < f(x_2)).

Determine the intervals on which (h(x) = x^3 - 3x) is increasing and decreasing.

For (f(x) = -x + 5), is it increasing or decreasing? Verify with two test values.