Lecture 24: Concavity & Curve Sketching

Point of reflection - curve changes from one concavity to another

Curve sketching

1. critical values from first derivative

Example

$f (x) = x2 e2$ $f^1 (x) = 2x e e + x2 ex$

at x = 0 , -2 critical points

2. Intervals ↑ or ↓ around CP

3. Local extrema values

4. Concavity using second derivative

$f^11 (r) = 2 ex + 2 x ex + 2 x e x$

= $ex ( x2 + 4 x + 2 )$

$f^11 (-4 )$ = +ve up!

$f^11 (-2)$ = - ve down !

$f^11 (0)$ = +ve up!

5. Sketch

Second derivative test

$f^11 (c) = 0$ & $f^1 (c) > 0$ local min at c

$f^11 (c) = 0$ & $f^1 (c) < 0$ local max at c