vault backup: 2025-02-25 09:48:29

education/math/MATH1210 (calc 1)/Max and Min.md

@@ -11,4 +11,14 @@ The absolute **minimum** is the smallest possible output value for a function.
 A number is considered critical if the output of a function exists and $\dfrac{d}{dx}$ is zero or undefined.
 
 # Local Max/Min
-A local max/min is a peak or trough at any point along the graph.
+A local max/min is a peak or trough at any point along the graph.
+
+# Extreme Value Theorem
+If $f$ is a continuous function in a closed interval $[a, b]$, then $f$ achieves both an absolute maximum and an absolute minimum in $[a, b]$. Furthermore, the absolute extrema occur at $a$ or at $b$ or at a critical number between $a$ and $b$.
+
+## Example
+> Find the absolute maximum and absolute minimum of the function $f(x) = x^2 -3x + 2$ on the closed interval $[0, 2]$:
+1. $x=0$ and $x=2$ are both critical numbers because they are endpoints. Endpoints are *always* critical numbers because $\dfrac{d}{dx}$ is undefined.
+2. $\dfrac{d}{dx} x^2 -3x + 2 = 2x -3$
+3. Setting the derivative to zero, $0 = 2x - 3$
+4. Solving for x, we get $x = \dfrac{3}{2}$. Three halves is a critical number because $