Q.Find the intervals in which the function $y = \dfrac{\ln x}{x}$ is increasing and decreasing.

Monotonic Function Analysis

A function is monotonic on an interval when it moves in a single direction across that interval — either always rising or always falling, never doubling back. Derivatives give us a clean, mechanical way to detect this, which is why monotonicity is one of the first applications of the derivative.

Increasing, Decreasing, Monotonic

On an interval $I$, a function $f$ is:

• increasing if $x_1 < x_2 \Rightarrow f(x_1) \le f(x_2)$,
• strictly increasing if $x_1 < x_2 \Rightarrow f(x_1) < f(x_2)$,
• decreasing if $x_1 < x_2 \Rightarrow f(x_1) \ge f(x_2)$,
• strictly decreasing if $x_1 < x_2 \Rightarrow f(x_1) > f(x_2)$.

A function that is either increasing throughout $I$ or decreasing throughout $I$ is called monotonic on $I$.

The Derivative Test

The slope of the tangent tells you the direction of travel. If $f$ is differentiable on an open interval $I$:

The idea is intuitive: a positive slope means the graph climbs as you move right, a negative slope means it falls.

How to Analyse Monotonicity

1. Compute $f'(x)$.
2. Solve $f'(x) = 0$ (and note where $f'$ is undefined). These critical points split the domain into intervals.
3. Test the sign of $f'$ in each interval.
4. Read off where $f$ increases ($f' > 0$) and decreases ($f' < 0$).

Example. For $f(x) = x^2 - 4x + 1$, $f'(x) = 2x - 4$. So $f'(x) < 0$ for $x < 2$ and $f'(x) > 0$ for $x > 2$: the function decreases on $(-\infty, 2)$ and increases on $(2, \infty)$. …