Complete function study: the 6-step method

The 6-step ritual

Step 1 — Domain

Forbidden values:

• Denominator = 0
• Negative or 0 argument for $\sqrt{}$, $\ln$
• $\tan$ at $\pi/2 + k\pi$

Step 2 — Limits at boundaries

Every boundary of the domain, including $\pm\infty$.

Step 3 — Derivative

Compute $f'(x)$.

Step 4 — Sign of derivative

• $f' > 0$ ⇒ increasing
• $f' < 0$ ⇒ decreasing
• $f' = 0$ isolated ⇒ extremum

Step 5 — Monotonicity table

Three rows: $x$, $f'(x)$, $f(x)$, with limits at boundaries and extrema values.

Step 6 — Curve

Draw:

• Notable points (extrema, intersections)
• Vertical asymptotes
• Horizontal asymptotes

Quick example: $f(x) = x^3 - 3x$

Domain: $\mathbb{R}$.

Limits: $\pm\infty$.

Derivative: $f'(x) = 3(x-1)(x+1)$.

Sign:

• $f' > 0$ on $(-\infty, -1) \cup (1, +\infty)$
• $f' < 0$ on $(-1, 1)$

Table:

• Increasing on $(-\infty, -1]$
• Decreasing on $[-1, 1]$
• Increasing on $[1, +\infty)$
• Local max at $-1$: $f(-1) = 2$
• Local min at $1$: $f(1) = -2$

Exam tips

• Always state the domain first.
• Justify each sign change of $f'$.
• Name asymptotes: “vertical asymptote $x = a$”.

Common mistakes

1. Forgetting the domain.
2. Confusing local and global extrema.
3. Wrong $\pm\infty$ placement.

?À toi de jouer

In a variation table, if f'(x) > 0 on an interval, f is:

A.Strictly increasingB.Strictly decreasingC.ConstantD.Undefined

Valider