From f1e3f77c60272725f17166ff66792ed41a93ac7c Mon Sep 17 00:00:00 2001 From: eggy Date: Mon, 16 Nov 2020 17:02:00 -0500 Subject: [PATCH] math: add extrema --- docs/mhf4u7.md | 26 +++++++++++++++++++++++++- 1 file changed, 25 insertions(+), 1 deletion(-) diff --git a/docs/mhf4u7.md b/docs/mhf4u7.md index 4a63059..9a8d1e7 100644 --- a/docs/mhf4u7.md +++ b/docs/mhf4u7.md @@ -552,7 +552,7 @@ When solving for questions that ask for rate of change related to other rates of - If $f´(x) = 0$ in the interval $[a,b]$, $f$ is **constant** on $[a,b]$. - The points where $f´(x)=0$ are the **critical**/maximum/minimum points. -Function only change whether they are increasing/decreasing/constant at the **critical points**/"relative extrema". +Functions only change whether they are increasing/decreasing/constant at the **critical points**/"relative extrema". These points and whether the intervals between them increase/decrease can be found by using an **interval chart/line** using the first derivative. @@ -562,6 +562,30 @@ These points and whether the intervals between them increase/decrease can be fou - $f$ is decreasing on $(-∞, -3) \cup (-3, 0) \cup (3, ∞)$. - $f$ is increasing on $(0, 1) \cup (1, 3)$. +### Extrema + +Extrema are the maximum and minimum points in a function or an interval of a function. They must be at **critical points**—where $f(x)=0$ or $f(x)=\text{DNE}$, and may include the **boundary points** if looking for extrema in a given interval. + +The highest and lowest point(s) of $f(x)$ are known as the **absolute** maximum/minimum of $f(x)$. + +Any other **relative/local** maxima or minima are such that all of the points around that point are higher or lower. + +**Fermat's theorem** states that if $f(c)$ is a local extremum, $c$ must be a critical number of $f$. Therefore, if $f$ is continuous in the closed interval $[a,b]$, the absolute extrema of $f$ must occur at $a$, $b$, or a critical number. + +To find the extrema of a **continuous** function $f(x)$, where $x=a$ is a critical value, the **first derivative test** may be used with the assistance of an interval chart/line. If only an interval of a function is under consideration, the boundary points must be taken under consideration as well. + + - If $f´(a)$ changes from positive to negative, there is a relative/local minimum at $x=a$. + - If $f´(a)$ changes from negative to positive, there is a relative/local maximum at $x=a$. + - If the sign is the same on both sides, there is no extrema at $x=a$. + - The greatest/least relative/local maximum/minimum is the absolute maximum/minimum. + +!!! example + The absolute minimum of $f(x)=x^2$ is at $(0,0)$. There is no absolute maximum nor are there any other relative/local maximum/minimum points. + +!!! warning + - There can be multiple absolute maxima/minima if there are multiple points that are both highest/lowest. + - If a function is a horizontal line, the absolute maximum and minimum for $x \in \text{domain} is $y$. + ## Resources - [IB Math Analysis and Approaches Syllabus](/resources/g11/ib-math-syllabus.pdf)