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Monotonicity

Tags

Calculus

Cegep/1

Word count

392 words

Reading time

3 minutes

Property of a function over an interval $I$ where $\mathrm{\forall }x\in I,f^{\prime }(x)>0$ (increasing) or $f^{\prime }(x)<0$ (decreasing)

Evaluation

To find the intervals of monotonicity of $f$:

1. Find the domain of $f$.
2. Find and simplify $f^{\prime }$.
3. Find all [[critical point|critical points]] of $f$.
4. Construct a sign table with the critical points.

Examples

Find the intervals of monotonicity, critical points and local extrema of $f(x)=\frac{x^2+3}{x-1}$.

$f(x)$ is undefined when $x-1\ne 0⟹x\ne 1$. So, the domain of $f$ is $x\in \mathbb{R}\setminus \{1\}$.

$$\begin{array}{rl}{f}^{\prime }(x)& =\frac{\mathrm{d}}{\mathrm{d}x}\frac{x^2+3}{x-1}\\ & =\frac{(x-1)2x-(x^2+3)}{(x-1{)}^2}\\ & =\frac{(x-3)(x+1)}{(x-1{)}^2}\\ ⟹0& =x-3\text{ or }0=x+1\\ x& =3\text{ or }x=-1\end{array}$$

$x=-1$ and $x=3$ are critical points. Since $x=1\notin dom(f)$, it cannot be a critical point.

$-\mathrm{\infty }$		-1		1		3		$\mathrm{\infty }$
$f^{\prime }(x)$	+		-		-		+
$f(x)$	inc		dec		dec		inc

$f$ has a relative maximum at $x=-1$ and a relative minimum at $x=3$.
$x=1$ cannot be a local extremum since $x=1\notin dom(f)$.

---

Find the intervals of monotonicity, critical points and local extrema of

$$f(x)=x^2(x^2-1{)}^{1/3}$$

given

$$f^{\prime }(x)=\frac{2x(4x^2-3)}{3(x^2-1{)}^{2/3}}$$

$f^{\prime }(x)$ is defined $\mathrm{\forall }x\in \mathbb{R}$.

$$\begin{array}{rl}{f}^{\prime }(x)& =\frac{2x(4x^2-3)}{3(x^2-1{)}^{2/3}}\\ ⟹0& =2x\text{ or }0=4x^2-3\\ x& =0\text{ or }\pm \frac{\sqrt{3}}{2}\end{array}$$

$f^{\prime }(x)$ is undefined when $3(x^2-1{)}^{2/3}=0⟹x=\pm 1$.
$x=0$, $x=\pm \frac{\sqrt{3}}{2}$ and $x=\pm 1$ are critical points.

$-\mathrm{\infty }$		-1		$-\frac{\sqrt{3}}{2}$		0		$\frac{\sqrt{3}}{2}$		1		$\mathrm{\infty }$
$f^{\prime }(x)$	-		-		+		-		+		+
$f(x)$	dec		dec		inc		dec		inc		inc

At $x=\pm 1$, this is no extremum since $f^{\prime }$ does not change sign.
At $x=\pm \frac{\sqrt{3}}{2}$, $f$ has a local minimum, and at $x=0$ there is a local maximum.

Contributors

Ajitani Hifumi

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