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)=x2e2 f (x) = x2 e2 f1(x)=2xee+x2ex 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

f11(r)=2ex+2xex+2xex f^11 (r) = 2 ex + 2 x ex + 2 x e x

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

f11(4)f^11 (-4 ) = +ve up!

f11(2) f^11 (-2)  = - ve down !

f11(0) f^11 (0)  = +ve up!

5. Sketch

Second derivative test

f11(c)=0f^11 (c) = 0  & f1(c)>0f^1 (c) > 0 local min at c

f11(c)=0f^11 (c) = 0 & f1(c)<0 f^1 (c) < 0 local max at c


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