The The Mean Value Theorem has an everyday consequence: the sign of the derivative tells where a function rises and falls. That turns finding peaks and valleys into reading a sign chart.
Key idea
If $f' > 0$ on an interval, $f$ is increasing there; if $f' < 0$, decreasing. First derivative test: at a critical number $c$, if $f'$ changes from $+$ to $-$, $f$ has a local max; from $-$ to $+$, a local min; if $f'$ does not change sign, neither.
Why it works
Take $x_1 < x_2$ in an interval where $f' > 0$. By the MVT, $f(x_2) - f(x_1) = f'(c)(x_2 - x_1)$ for some $c$ between them. Both factors are positive, so $f(x_2) > f(x_1)$: the function increases. The test follows: rising then falling makes a peak; falling then rising makes a valley.
Making a sign chart
- Find the critical numbers and mark them on a number line.
- Pick a test point in each piece and find the sign of $f'$ there (or use the signs of the factors).
- Read off increasing and decreasing intervals, and classify each critical number.
Worked example: A cubic
Find where $f(x) = \frac{x^3}{3} - x^2 - 3x$ increases and decreases, and classify its critical numbers.
- $f'(x) = x^2 - 2x - 3 = (x + 1)(x - 3)$: critical numbers $-1$ and $3$.
- Test points: $f'(-2) = 5 > 0$, $f'(0) = -3 < 0$, $f'(4) = 5 > 0$.
- So $f$ increases on $(-\infty, -1)$ and $(3, \infty)$ and decreases on $(-1, 3)$.
- $+ \to -$ at $-1$: local max. $- \to +$ at 3: local min.
Answer: Local max at $x = -1$, local min at $x = 3$.
Worked example: A flat point that is not an extreme
Classify the critical numbers of $f(x) = x^4 - 4x^3$.
- $f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3)$: critical numbers $0$ and $3$.
- $4x^2 \ge 0$, so the sign of $f'$ is the sign of $x - 3$: negative on both sides of 0, positive after 3.
- At 0 the sign does not change: neither max nor min. At 3: $- \to +$, a local min.
Answer: $x = 0$: neither; $x = 3$: local min, $f(3) = -27$.
Common mistake
Finding where $f' = 0$ and stopping there, calling every such point a max or min. The second example shows a flat point that is neither. Check: always test the sign of $f'$ on both sides of each critical number.