Increasing and Decreasing Intervals

A function is increasing wherever its derivative is positive and decreasing wherever its derivative is negative. The transitions happen exactly at the critical points (where $f'(x) = 0$). This visualization color-codes the curve green where it's increasing and red where it's decreasing — turning an algebraic question into a glance.

The Physics — Just Algebra

For $f(x) = x^{3} - 6x^{2} + 9x + 1$, take the derivative, factor it, and analyze the sign of $f'(x)$ in each interval determined by its roots.

Step-by-Step Solution

Given: $f(x) = x^{3} - 6x^{2} + 9x + 1$.

Find: The intervals where $f$ is increasing and decreasing.

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Step 1 — Compute the derivative.

$f'(x) = 3x^{2} - 12x + 9$

Step 2 — Factor the derivative.

Pull out the GCF of 3:

$f'(x) = 3(x^{2} - 4x + 3)$

Factor the quadratic. We need two numbers that multiply to 3 and add to $-4$ — those are $-1$ and $-3$:

$f'(x) = 3(x - 1)(x - 3)$

Step 3 — Find the critical points by setting $f'(x) = 0$.

$3(x - 1)(x - 3) = 0 \;\;\Longrightarrow\;\; x = 1 \text{ or } x = 3$

These two points divide the real line into three intervals: $(-\infty, 1)$, $(1, 3)$, $(3, \infty)$.

Step 4 — Test the sign of $f'$ in each interval.

Pick a test point inside each interval and evaluate the factored form $3(x - 1)(x - 3)$.

• Test $x = 0$ in $(-\infty, 1)$: $\;3(0 - 1)(0 - 3) = 3 \cdot (-1) \cdot (-3) = +9$ → positive → $f$ is increasing
• Test $x = 2$ in $(1, 3)$: $\;3(2 - 1)(2 - 3) = 3 \cdot (1) \cdot (-1) = -3$ → negative → $f$ is decreasing
• Test $x = 4$ in $(3, \infty)$: $\;3(4 - 1)(4 - 3) = 3 \cdot (3) \cdot (1) = +9$ → positive → $f$ is increasing

Step 5 — Identify local extrema.

The function changes from increasing to decreasing at $x = 1$, so $(1,\, f(1))$ is a local maximum. From decreasing to increasing at $x = 3$, so $(3,\, f(3))$ is a local minimum.

$f(1) = 1 - 6 + 9 + 1 = 5 \;\;\Longrightarrow\;\; \text{local max at } (1,\, 5)$

$f(3) = 27 - 54 + 27 + 1 = 1 \;\;\Longrightarrow\;\; \text{local min at } (3,\, 1)$

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Answer:

• $f$ is increasing on $\boxed{(-\infty,\, 1) \cup (3,\, \infty)}$
• $f$ is decreasing on $\boxed{(1,\, 3)}$
• Local maximum at $(1, 5)$, local minimum at $(3, 1)$

In the visualization the curve is drawn in green where it's increasing and red where it's decreasing, with the two critical points marked.

Try It

• Slide the point along the curve — the HUD changes from "INCREASING" (green) to "DECREASING" (red) as you cross the critical points.
• The curve is drawn in green for $x < 1$ and $x > 3$, and red between 1 and 3.
• The two critical points are marked with green stars. Slide the point onto them — the tangent line goes flat.
• Toggle show $f'(x)$ to see the derivative curve below: it sits above the $x$-axis where $f$ is increasing and below where $f$ is decreasing.