User:Annlex/Mathproject

CURVE SKETCHING

f(x)=x4+4x3+6x2+4x

-Roots:

0=x4+4x3+6x2+4x

0=x(x3+4x2+6x+4)

x1=0 x3+4x2+6x+4=0 x2=-2 (erraten)

Polynomdivision:

(x3+4x2+6x+4):(x+2)= x2+2x+2 -(x3+2x2)

2x2+6x -(2x2+4x) 2x+4 -(2x+4) 000

x2+2x+2=0 x1,2=

Roots:NS1(0/0),NS2(-2/0)

-1. Derivation:

f´(x)=4x3+12x2+12x+4

0=4x3+12x2+12x+4

x1= - 1 (erraten)

(4x3+12x2+12x+4) : (x+1) = 4x2+8x+4 -(4x3+4x2) -    8x2+12x -(8x2+8x) 4x+4 -(4x+4) ---         000

quadratic equation:

4x2+8x+4=0

4(x2+2x+1)=0

4(x+1)2=0

x1=x2=x3=-1 (tripple root)

-2. Derivation:

f´´(x)=12x+24x+12

f´´(-1)=12-24+12=0 (no solution)

f´(0)=4  (positiv gradient)

-TIP(-1/-1) (absolute minimum)

-Wf=[-1;+unendlich[

-Df=[-unendlich;+unentlich[

-There is no symmrtry to the koordinate plane.

general procedure:

f(x):

-roots: find out when the y-value is zero

f(x)=0