From e9e8d8f1de9895eccf02071174828c4f0bd56b73 Mon Sep 17 00:00:00 2001 From: eggy Date: Tue, 17 Nov 2020 17:39:59 -0500 Subject: [PATCH] math: concavity --- docs/mhf4u7.md | 10 ++++++++++ 1 file changed, 10 insertions(+) diff --git a/docs/mhf4u7.md b/docs/mhf4u7.md index 9a8d1e7..8c03209 100644 --- a/docs/mhf4u7.md +++ b/docs/mhf4u7.md @@ -586,6 +586,16 @@ To find the extrema of a **continuous** function $f(x)$, where $x=a$ is a critic - 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 $y$. +### Concavity + +!!! definition + A **point of inflection** on a curve is such that $f´´(x)=0 \text{ or DNE}$ and the signs around the point change (e.g., positive to negative). + + - An interval is **concave up** if it increases from left to right and tangent lines are drawn below the curve, so $f´´(x)>0$. It is shaped like a smile. + - An interval is **concave down** if it increases from left to right and tangent lines are drawn **above** the curve, and $f´´(x)<0$. It is shaped like a frown. + +Changes in concavity only occur at points of inflection, and their specific points can be identified using the **second derivative test**, which follows many of the same steps as the first derivative test but with the second derivative. + ## Resources - [IB Math Analysis and Approaches Syllabus](/resources/g11/ib-math-syllabus.pdf)