Increasing and Decreasing Functions — First Derivative Test Interpretation

For a differentiable function $f(x)$ on an interval:

• $f'(x) > 0$ for all $x$ in the interval $\implies$ $f$ is strictly increasing
• $f'(x) < 0$ for all $x$ in the interval $\implies$ $f$ is strictly decreasing
• $f'(x) = 0$ for all $x$ in the interval $\implies$ $f$ is constant

The derivative tells us the “direction” of the function at each point.

Set $f'(x) = 0$ and solve. The solutions are critical points — these are where the function might switch from increasing to decreasing (or vice versa).

Example: $f(x) = x^3 - 6x^2 + 9x + 1$

Critical points: $x = 1$ and $x = 3$.

The critical points divide the number line into intervals. Pick a test point in each:

So $f$ is increasing on $(-\infty, 1) \cup (3, \infty)$ and decreasing on $(1, 3)$.

graph TD
    A[Given f of x] --> B[Compute f prime of x]
    B --> C[Solve f prime of x = 0 for critical points]
    C --> D[Divide number line into intervals]
    D --> E[Pick test point in each interval]
    E --> F[Evaluate sign of f prime at test point]
    F --> G{f prime positive?}
    G -->|Yes| H[Function increasing on that interval]
    G -->|No| I[Function decreasing on that interval]