# §1.4 Calculus of One Variable

## §1.4(i) Monotonicity

If $f(x_{1})\leq f(x_{2})$ for every pair $x_{1}$, $x_{2}$ in an interval $I$ such that $x_{1}, then $f(x)$ is nondecreasing on $I$. If the $\leq$ sign is replaced by $<$, then $f(x)$ is increasing (also called strictly increasing) on $I$. Similarly for nonincreasing and decreasing (strictly decreasing) functions. Each of the preceding four cases is classified as monotonic; sometimes strictly monotonic is used for the strictly increasing or strictly decreasing cases.

## §1.4(ii) Continuity

A function $f(x)$ is continuous on the right (or from above) at $x=c$ if

 1.4.1 $f(c+)\equiv\lim_{x\to c+}f(x)=f(c),$ ⓘ Symbols: $\equiv$: equals by definition Referenced by: §1.4(ii) Permalink: http://dlmf.nist.gov/1.4.E1 Encodings: TeX, pMML, png See also: Annotations for §1.4(ii), §1.4 and Ch.1

that is, for every arbitrarily small positive constant $\epsilon$ there exists $\delta$ ($>0$) such that

 1.4.2 $|f(c+\alpha)-f(c)|<\epsilon,$ ⓘ Permalink: http://dlmf.nist.gov/1.4.E2 Encodings: TeX, pMML, png See also: Annotations for §1.4(ii), §1.4 and Ch.1

for all $\alpha$ such that $0\leq\alpha<\delta$. Similarly, it is continuous on the left (or from below) at $x=c$ if

 1.4.3 $f(c-)\equiv\lim_{x\to c-}f(x)=f(c).$ ⓘ Symbols: $\equiv$: equals by definition Referenced by: §1.4(ii) Permalink: http://dlmf.nist.gov/1.4.E3 Encodings: TeX, pMML, png See also: Annotations for §1.4(ii), §1.4 and Ch.1

And $f(x)$ is continuous at $c$ when both (1.4.1) and (1.4.3) apply.

If $f(x)$ is continuous at each point $c\in(a,b)$, then $f(x)$ is continuous on the interval $(a,b)$ and we write $f\in C(a,b)$. If also $f(x)$ is continuous on the right at $x=a$, and continuous on the left at $x=b$, then $f(x)$ is continuous on the interval $[a,b]$, and we write $f(x)\in C[a,b]$.

A removable singularity of $f(x)$ at $x=c$ occurs when $f(c+)=f(c-)$ but $f(c)$ is undefined. For example, $f(x)=(\sin x)/x$ with $c=0$.

A simple discontinuity of $f(x)$ at $x=c$ occurs when $f(c+)$ and $f(c-)$ exist, but $f(c+)\not=f(c-)$. If $f(x)$ is continuous on an interval $I$ save for a finite number of simple discontinuities, then $f(x)$ is piecewise (or sectionally) continuous on $I$. For an example, see Figure 1.4.1

## §1.4(iii) Derivatives

The derivative $f^{\prime}(x)$ of $f(x)$ is defined by

 1.4.4 $f^{\prime}(x)=\frac{\mathrm{d}f}{\mathrm{d}x}=\lim_{h\to 0}\frac{f(x+h)-f(x)}{% h}.$ ⓘ Defines: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative Referenced by: §1.8(ii) Permalink: http://dlmf.nist.gov/1.4.E4 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4 and Ch.1

When this limit exists $f$ is differentiable at $x$.

 1.4.5 $(f+g)^{\prime}(x)=f^{\prime}(x)+g^{\prime}(x),$ ⓘ A&S Ref: 3.3.2 Permalink: http://dlmf.nist.gov/1.4.E5 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4 and Ch.1
 1.4.6 $(fg)^{\prime}(x)=f^{\prime}(x)g(x)+f(x)g^{\prime}(x),$ ⓘ A&S Ref: 3.3.3 Permalink: http://dlmf.nist.gov/1.4.E6 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4 and Ch.1
 1.4.7 $\left(\frac{f}{g}\right)^{\prime}(x)=\frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)% }{(g(x))^{2}}.$ ⓘ A&S Ref: 3.3.4 Permalink: http://dlmf.nist.gov/1.4.E7 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4 and Ch.1

### Higher Derivatives

 1.4.8 $f^{(2)}(x)=\frac{{\mathrm{d}}^{2}f}{{\mathrm{d}x}^{2}}=\frac{\mathrm{d}}{% \mathrm{d}x}\left(\frac{\mathrm{d}f}{\mathrm{d}x}\right),$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative Permalink: http://dlmf.nist.gov/1.4.E8 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1
 1.4.9 $f^{(n)}=f^{(n)}(x)=\frac{\mathrm{d}}{\mathrm{d}x}f^{(n-1)}(x).$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.4.E9 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

If $f^{(n)}$ exists and is continuous on an interval $I$, then we write $f\in C^{n}(I)$. When $n\geq 1$, $f$ is continuously differentiable on $I$. When $n$ is unbounded, $f$ is infinitely differentiable on $I$ and we write $f\in C^{\infty}(I)$.

### Chain Rule

For $h(x)=f(g(x))$,

 1.4.10 $h^{\prime}(x)=f^{\prime}(g(x))g^{\prime}(x).$ ⓘ A&S Ref: 3.3.5 Permalink: http://dlmf.nist.gov/1.4.E10 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

### Maxima and Minima

A necessary condition that a differentiable function $f(x)$ has a local maximum (minimum) at $x=c$, that is, $f(x)\leq f(c)$, ($f(x)\geq f(c)$) in a neighborhood $c-\delta\leq x\leq c+\delta$ ($\delta>0$) of $c$, is $f^{\prime}(c)=0$.

### Mean Value Theorem

If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists a point $c\in(a,b)$ such that

 1.4.11 $f(b)-f(a)=(b-a)f^{\prime}(c).$ ⓘ Permalink: http://dlmf.nist.gov/1.4.E11 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

If $f^{\prime}(x)\geq 0$ ($\leq 0$) ($=0$) for all $x\in(a,b)$, then $f$ is nondecreasing (nonincreasing) (constant) on $(a,b)$.

### Leibniz’s Formula

 1.4.12 $(fg)^{(n)}=f^{(n)}g+\genfrac{(}{)}{0.0pt}{}{n}{1}f^{(n-1)}g^{\prime}+\dots+% \genfrac{(}{)}{0.0pt}{}{n}{k}f^{(n-k)}g^{(k)}+\dots+fg^{(n)}.$ ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $k$: integer and $n$: nonnegative integer A&S Ref: 3.3.8 Permalink: http://dlmf.nist.gov/1.4.E12 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

### Faà Di Bruno’s Formula

 1.4.13 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}f(g(x))=\sum\left(\frac{n!}{m_{1}!m_% {2}!\cdots m_{n}!}\right)f^{(k)}(g(x))\*\left(\frac{g^{\prime}(x)}{1!}\right)^% {m_{1}}\left(\frac{g^{\prime\prime}(x)}{2!}\right)^{m_{2}}\dots\left(\frac{g^{% (n)}(x)}{n!}\right)^{m_{n}},$

where the sum is over all nonnegative integers $m_{1},m_{2},\dots,m_{n}$ that satisfy $m_{1}+2m_{2}+\dots+nm_{n}=n$, and $k=m_{1}+m_{2}+\dots+m_{n}$.

### L’Hôpital’s Rule

If

 1.4.14 $\lim\limits_{x\to a}f(x)=\lim\limits_{x\to a}g(x)=0\;\;\mbox{(or \infty)},$ ⓘ Permalink: http://dlmf.nist.gov/1.4.E14 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

then

 1.4.15 $\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f^{\prime}(x)}{g^{\prime}(x)}$ ⓘ A&S Ref: 3.4.1 Referenced by: §1.4(iii), Erratum (V1.1.11) for Subsection 1.4(iii) Permalink: http://dlmf.nist.gov/1.4.E15 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

when the last limit exists. We do assume that $g^{\prime}(x)\neq 0$ for all $x$ in some neighborhood of $a$ with $x\neq a$.

## §1.4(iv) Indefinite Integrals

If $F^{\prime}(x)=f(x)$, then $\int f\,\mathrm{d}x=F(x)+C$, where $C$ is a constant.

### Integration by Parts

 1.4.16 $\int fg\,\mathrm{d}x=\left(\int f\,\mathrm{d}x\right)g-\int\left(\int f\,% \mathrm{d}x\right)\frac{\mathrm{d}g}{\mathrm{d}x}\,\mathrm{d}x.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $\,\mathrm{d}\NVar{x}$: differential and $\int$: integral A&S Ref: 3.3.13 Permalink: http://dlmf.nist.gov/1.4.E16 Encodings: TeX, pMML, png See also: Annotations for §1.4(iv), §1.4(iv), §1.4 and Ch.1
 1.4.17 $\int x^{n}\,\mathrm{d}x=\begin{cases}\dfrac{x^{n+1}}{n+1}+C,&\quad n\not=-1,\\ \ln\left|x\right|+C,&\quad n=-1.\end{cases}$

For the function $\ln$ see §4.2(i).

See §§4.10, 4.26(ii), 4.26(iv), 4.40(ii), and 4.40(iv) for indefinite integrals involving the elementary functions.

For extensive tables of integrals, see Apelblat (1983), Bierens de Haan (1867), Gradshteyn and Ryzhik (2000), Gröbner and Hofreiter (1949, 1950), and Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).

## §1.4(v) Definite Integrals

Suppose $f(x)$ is defined on $[a,b]$. Let $a=x_{0}, and $\xi_{j}$ denote any point in $[x_{j},x_{j+1}]$, $j=0,1,\dots,n-1$. Then

 1.4.18 $\int^{b}_{a}f(x)\,\mathrm{d}x=\lim\sum^{n-1}_{j=0}f(\xi_{j})(x_{j+1}-x_{j})$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral, $j$: integer, $n$: nonnegative integer and $\xi_{\NVar{j}}$: point Permalink: http://dlmf.nist.gov/1.4.E18 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4 and Ch.1

as $\max(x_{j+1}-x_{j})\to 0$. Continuity, or piecewise continuity, of $f(x)$ on $[a,b]$ is sufficient for the limit to exist.

 1.4.19 $\int^{b}_{a}(cf(x)+dg(x))\,\mathrm{d}x=c\int^{b}_{a}f(x)\,\mathrm{d}x+d\int^{b% }_{a}g(x)\,\mathrm{d}x,$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/1.4.E19 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4 and Ch.1

$c$ and $d$ constants.

 1.4.20 $\int^{b}_{a}f(x)\,\mathrm{d}x=-\int^{a}_{b}f(x)\,\mathrm{d}x.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/1.4.E20 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4 and Ch.1
 1.4.21 $\int^{b}_{a}f(x)\,\mathrm{d}x=\int^{c}_{a}f(x)\,\mathrm{d}x+\int^{b}_{c}f(x)\,% \mathrm{d}x.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/1.4.E21 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4 and Ch.1

### Infinite Integrals

 1.4.22 $\int^{\infty}_{a}f(x)\,\mathrm{d}x=\lim_{b\to\infty}\int^{b}_{a}f(x)\,\mathrm{% d}x.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential and $\int$: integral Referenced by: §1.4(v), §1.5(v) Permalink: http://dlmf.nist.gov/1.4.E22 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

Similarly for $\int^{a}_{-\infty}$. Next, if $f(b)=\pm\infty$, then

 1.4.23 $\int^{b}_{a}f(x)\,\mathrm{d}x=\lim_{c\to b-}\int^{c}_{a}f(x)\,\mathrm{d}x.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential and $\int$: integral Referenced by: §1.4(v), §1.5(v) Permalink: http://dlmf.nist.gov/1.4.E23 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

Similarly when $f(a)=\pm\infty$.

When the limits in (1.4.22) and (1.4.23) exist, the integrals are said to be convergent. If the limits exist with $f(x)$ replaced by $|f(x)|$, then the integrals are absolutely convergent. Absolute convergence also implies convergence.

### Cauchy Principal Values

Let $c\in(a,b)$ and assume that $\int_{a}^{c-\epsilon}f(x)\,\mathrm{d}x$ and $\int_{c+\epsilon}^{b}f(x)\,\mathrm{d}x$ exist when $0<\epsilon<\min(c-a,b-c)$, but not necessarily when $\epsilon=0$. Then we define

 1.4.24 $\pvint^{b}_{a}f(x)\,\mathrm{d}x=P\int^{b}_{a}f(x)\,\mathrm{d}x=\lim_{\epsilon% \to 0+}\left(\int^{c-\epsilon}_{a}f(x)\,\mathrm{d}x+\int^{b}_{c+\epsilon}f(x)% \,\mathrm{d}x\right),$ ⓘ Defines: $\pvint_{\NVar{a}}^{\NVar{b}}$: Cauchy principal value Symbols: $\,\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/1.4.E24 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

when this limit exists.

Similarly, assume that $\int_{-b}^{b}f(x)\,\mathrm{d}x$ exists for all finite values of $b$ ($>0$), but not necessarily when $b=\infty$. Then we define

 1.4.25 $\pvint^{\infty}_{-\infty}f(x)\,\mathrm{d}x=P\int^{\infty}_{-\infty}f(x)\,% \mathrm{d}x=\lim_{b\to\infty}\int^{b}_{-b}f(x)\,\mathrm{d}x,$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral and $\pvint_{\NVar{a}}^{\NVar{b}}$: Cauchy principal value Referenced by: §1.14(i) Permalink: http://dlmf.nist.gov/1.4.E25 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

when this limit exists.

### Fundamental Theorem of Calculus

For $F^{\prime}(x)=f(x)$ with $f(x)$ continuous,

 1.4.26 $\int^{b}_{a}f(x)\,\mathrm{d}x=F(b)-F(a),$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/1.4.E26 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1
 1.4.27 $\frac{\mathrm{d}}{\mathrm{d}x}\int^{x}_{a}f(t)\,\mathrm{d}t=f(x).$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $\,\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/1.4.E27 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

### Change of Variables

If $\phi^{\prime}(x)$ is continuous or piecewise continuous, then

 1.4.28 $\int^{b}_{a}f(\phi(x))\phi^{\prime}(x)\,\mathrm{d}x=\int^{\phi(b)}_{\phi(a)}f(% t)\,\mathrm{d}t.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral and $\phi(\NVar{x})$: function Permalink: http://dlmf.nist.gov/1.4.E28 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

### First Mean Value Theorem

For $f(x)$ continuous and $\phi(x)\geq 0$ and integrable on $[a,b]$, there exists $c\in[a,b]$, such that

 1.4.29 $\int^{b}_{a}f(x)\phi(x)\,\mathrm{d}x=f(c)\int^{b}_{a}\phi(x)\,\mathrm{d}x.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral and $\phi(\NVar{x})$: function Permalink: http://dlmf.nist.gov/1.4.E29 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

### Second Mean Value Theorem

For $f(x)$ monotonic and $\phi(x)$ integrable on $[a,b]$, there exists $c\in[a,b]$, such that

 1.4.30 $\int^{b}_{a}f(x)\phi(x)\,\mathrm{d}x=f(a)\int^{c}_{a}\phi(x)\,\mathrm{d}x+f(b)% \int^{b}_{c}\phi(x)\,\mathrm{d}x.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral and $\phi(\NVar{x})$: function Permalink: http://dlmf.nist.gov/1.4.E30 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

### Repeated Integrals

If $f(x)$ is continuous or piecewise continuous on $[a,b]$, then

 1.4.31 $\int_{a}^{b}\,\mathrm{d}x_{n}\int_{a}^{x_{n}}\,\mathrm{d}x_{n-1}\cdots\int_{a}% ^{x_{2}}\,\mathrm{d}x_{1}\int_{a}^{x_{1}}f(x)\,\mathrm{d}x=\frac{1}{n!}\int_{a% }^{b}(b-x)^{n}f(x)\,\mathrm{d}x.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential, $!$: factorial (as in $n!$), $\int$: integral and $n$: nonnegative integer Referenced by: §1.15(vi), §1.4(v) Permalink: http://dlmf.nist.gov/1.4.E31 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

### Square-Integrable Functions

A function $f(x)$ is square-integrable if

 1.4.32 $\|f\|^{2}_{2}\equiv\int^{b}_{a}|f(x)|^{2}\,\mathrm{d}x<\infty.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\equiv$: equals by definition and $\int$: integral Permalink: http://dlmf.nist.gov/1.4.E32 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

### Functions of Bounded Variation

With $a, the total variation of $f(x)$ on a finite or infinite interval $(a,b)$ is

 1.4.33 $\mathcal{V}_{a,b}\left(f\right)=\sup\sum^{n}_{j=1}|f(x_{j})-f(x_{j-1})|,$ ⓘ Defines: $\mathcal{V}\left(\NVar{f}\right)$: total variation and $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$: total variation Symbols: $\sup$: least upper bound (supremum), $j$: integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.4.E33 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

where the supremum is over all sets of points $x_{0} in the closure of $(a,b)$, that is, $(a,b)$ with $a,b$ added when they are finite. If $\mathcal{V}_{a,b}\left(f\right)<\infty$, then $f(x)$ is of bounded variation on $(a,b)$. In this case, $g(x)=\mathcal{V}_{a,x}\left(f\right)$ and $h(x)=\mathcal{V}_{a,x}\left(f\right)-f(x)$ are nondecreasing bounded functions and $f(x)=g(x)-h(x)$.

If $f(x)$ is continuous on the closure of $(a,b)$ and $f^{\prime}(x)$ is continuous on $(a,b)$, then

 1.4.34 $\mathcal{V}_{a,b}\left(f\right)=\int^{b}_{a}\left|f^{\prime}(x)\right|\,% \mathrm{d}x,$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral and $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$: total variation Referenced by: §1.4(v), §1.4(v), Erratum (V1.0.28) for Equation (1.4.34) Permalink: http://dlmf.nist.gov/1.4.E34 Encodings: TeX, pMML, png Correction (effective with 1.0.28): The integrand was corrected so that the absolute value does not include the differential. Suggested 2020-08-11 by Tran Quoc Viet See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

whenever this integral exists.

Lastly, whether or not the real numbers $a$ and $b$ satisfy $a, and whether or not they are finite, we define $\mathcal{V}_{a,b}\left(f\right)$ by (1.4.34) whenever this integral exists. This definition also applies when $f(x)$ is a complex function of the real variable $x$. For further information on total variation see Olver (1997b, pp. 27–29).

## §1.4(vi) Taylor’s Theorem for Real Variables

If $f(x)\in C^{n+1}[a,b]$, then

 1.4.35 $f(x)=\sum^{n}_{k=0}\frac{f^{(k)}(a)}{k!}(x-a)^{k}+R_{n},$ ⓘ Symbols: $!$: factorial (as in $n!$), $k$: integer, $n$: nonnegative integer and $R_{\NVar{n}}$: remainder A&S Ref: 3.6.4 Permalink: http://dlmf.nist.gov/1.4.E35 Encodings: TeX, pMML, png See also: Annotations for §1.4(vi), §1.4 and Ch.1
 1.4.36 $R_{n}=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1},$ $a, ⓘ Symbols: $!$: factorial (as in $n!$), $n$: nonnegative integer and $R_{\NVar{n}}$: remainder A&S Ref: 3.6.5 Permalink: http://dlmf.nist.gov/1.4.E36 Encodings: TeX, pMML, png See also: Annotations for §1.4(vi), §1.4 and Ch.1

and

 1.4.37 $R_{n}=\frac{1}{n!}\int^{x}_{a}(x-t)^{n}f^{(n+1)}(t)\,\mathrm{d}t.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential, $!$: factorial (as in $n!$), $\int$: integral, $n$: nonnegative integer and $R_{\NVar{n}}$: remainder A&S Ref: 3.6.3 Permalink: http://dlmf.nist.gov/1.4.E37 Encodings: TeX, pMML, png See also: Annotations for §1.4(vi), §1.4 and Ch.1

## §1.4(vii) Maxima and Minima

If $f(x)$ is twice-differentiable, and if also $f^{\prime}(x_{0})=0$ and $f^{\prime\prime}(x_{0})<0$ ($>0$), then $x=x_{0}$ is a local maximum (minimum) (§1.4(iii)) of $f(x)$. The overall maximum (minimum) of $f(x)$ on $[a,b]$ will either be at a local maximum (minimum) or at one of the end points $a$ or $b$.

## §1.4(viii) Convex Functions

A function $f(x)$ is convex on $(a,b)$ if

 1.4.38 $f((1-t)c+td)\leq(1-t)f(c)+tf(d)$ ⓘ Permalink: http://dlmf.nist.gov/1.4.E38 Encodings: TeX, pMML, png See also: Annotations for §1.4(viii), §1.4 and Ch.1

for any $c,d\in(a,b)$, and $t\in[0,1]$. See Figure 1.4.2. A similar definition applies to closed intervals $[a,b]$.

If $f(x)$ is twice differentiable, then $f(x)$ is convex iff $f^{\prime\prime}(x)\geq 0$ on $(a,b)$. A continuously differentiable function is convex iff the curve does not lie below its tangent at any point.