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
x_1,2=

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)

-W_f=[-1;+unendlich[
-D_f=[-unendlich;+unentlich[

-There is no symmrtry to the koordinate plane.

• 

  Caption1

general procedure:

f(x):

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

f(x)=0