Fundamental Theorem of Calculus (FTC)Prerequisite KnowledgeCalculus IRelative Extrema*First Derivative Test* for Local Maxima*First Derivative Test* for Local MinimaIntervals of IncreaseIntervals of DecreaseConcavityPracticeCC BY-NC-SA 4.0

**Author**: John J Weber III, PhD
**Corresponding Textbook Sections**:

**Section 4.9**– Antiderivatives**Section 5.3**– The Fundamental Theorem of Calculus**Section 5.4**– Indefinite Integrals and the Net Change Theorem

**NOTE**: One way to use the information contained within this document is to *graphically* **verify** antiderivatives.

**Definition**: A function **local (relative) maximum** at

**Definition**: A function **local (relative) minimum** at

**Definition**: Critical numbers are location(s), i.e.,

How to find locations, called

**critical numbers**, of local (relative) extrema:*Analytically*:- Solve
solve for${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=0$ -values;$x$ **and** - find
-values such that$x$ .${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=\text{dne}$

- Solve
*Graphically*:Find

-values where the graph of$x$ touches or crosses the${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)$ -axis;$x$ **and**find

-values where$x$ , i.e., where the graph of${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=\text{dne}$ has${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)$ - a discontinuity; or
- a cusp; or
- a vertical tangent line.

**NOTE**: A critical number does not always refer to a local extremum.

*Analytically*: has a local maximum at$f(x)$ if$x={x}_{0}$ for all${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)>0$ in some interval$x$ ;$(a,{x}_{0})$ **and** for all${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)<0$ in some interval$x$ .$({x}_{0},b)$

*Graphically*: has a local maximum at$f(x)$ if$x={x}_{0}$ - the graph of
is${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)$ **above**the -axis for all$x$ in some interval$x$ ;$(a,{x}_{0})$ **and** - the graph of
is${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)$ **below**the -axis for all$x$ in some interval$x$ .$({x}_{0},b)$

- the graph of

*Analytically*: has a local maximum at$f(x)$ if$x={x}_{0}$ for all${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)>0$ in some interval$x$ ;$(a,{x}_{0})$ **and** for all${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)<0$ in some interval$x$ .$({x}_{0},b)$

*Graphically*: has a local minimum at$f(x)$ if$x={x}_{0}$ - the graph of
is${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)$ **below**the -axis for all$x$ in some interval$x$ ;$(a,{x}_{0})$ **and** - the graph of
is${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)$ **above**the -axis for all$x$ in some interval$x$ .$({x}_{0},b)$

- the graph of

*Analytically*: is increasing at a point$f(x)$ if and only if there exists some interval${x}_{0}$ containing$(a,b)$ such that${x}_{0}$ for all$f({x}_{0})>f(x)$ in$x$ to the left of$(a,b)$ and${x}_{0}$ for all$f({x}_{0})<f(x)$ in$x$ to the right of$(a,b)$ ;${x}_{0}$ **or** is increasing at a point$f(x)$ if and only if${x}_{0}$ .${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}({x}_{0})>0$

*Graphically*: is increasing at a point$f(x)$ if and only if the graph of${x}_{0}$ is${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}({x}_{0})$ **above**the -axis.$x$

*Analytically*: is decreasing at a point$f(x)$ if and only if there exists some interval${x}_{0}$ containing$(a,b)$ such that${x}_{0}$ for all$f({x}_{0})<f(x)$ in$x$ to the left of$(a,b)$ and${x}_{0}$ for all$f({x}_{0})>f(x)$ in$x$ to the right of$(a,b)$ ;${x}_{0}$ **or** is decreasing at a point$f(x)$ if and only if${x}_{0}$ .${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}({x}_{0})<0$

*Graphically*: is decreasing at a point$f(x)$ if and only if the graph of${x}_{0}$ is${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}({x}_{0})$ **below**the -axis.$x$

A function

is$f(x)$ **concave up**at if and only if${x}_{0}$ is increasing at${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)$ .${x}_{0}$ - If
exists at${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime \prime}(x)$ and is positive, then${x}_{0}$ is$f(x)$ **concave up**at .${x}_{0}$

- If
A function

is$f(x)$ **concave down**at if and only if${x}_{0}$ is decreasing at${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)$ .${x}_{0}$ - If
exists at${f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime \prime}(x)$ and is negative, then${x}_{0}$ is$f(x)$ **concave down**at .${x}_{0}$

- If

**Example 01**:

- Find locations of all local extrema of
, if exists.$h(x)$ - Find all local extrema of
, if exists.$h(x)$ - Find interval(s) on which
is increasing, if exists.$h(x)$ - Find interval(s) on which
is decreasing, if exists.$h(x)$ - Find all inflection points of
, if exists.$h(x)$ - Find interval(s) on which
is concave up, if exists.$h(x)$ - Find interval(s) on which
is concave down, if exists.$h(x)$

**Example 02**:

- Find locations of all local extrema of
, if exists.$f(x)$ - Find all local extrema of
, if exists.$f(x)$ - Find interval(s) on which
is increasing, if exists.$f(x)$ - Find interval(s) on which
is decreasing, if exists.$f(x)$ - Find all inflection points of
, if exists.$f(x)$ - Find interval(s) on which
is concave up, if exists.$f(x)$ - Find interval(s) on which
is concave down, if exists.$f(x)$

**Example 03**:

- Find locations of all local extrema of
, if exists.$g(x)$ - Find all local extrema of
, if exists.$g(x)$ - Find locations of all absolute extrema of
, if exists.$g(x)$ - Find all absolute extrema. How do you know
has absolute extrema?$g(x)$ - Find interval(s) on which
is increasing, if exists.$g(x)$ - Find interval(s) on which
is decreasing, if exists.$g(x)$ - Find all inflection points of
, if exists.$g(x)$ - Find interval(s) on which
is concave up, if exists.$g(x)$ - Find interval(s) on which
is concave down, if exists.$g(x)$

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

**Created**: Monday, 18 January 2021 09:33 EDT
**Last Modified**: Monday, 15 August 2022 - 15:08 (EDT)