Find the first and second derivatives of S(x)=2x^3-18x^2+48x+220 and then create a sign diagram for both?  I'm having trouble with factoring the first derivative

embizze | High School Teacher | (Level 1) Educator Emeritus

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Given `S(x)=2x^3-18x^2+48x+220`

(a) `S'(x)=6x^2-36x+48`

`S'(x)=0 ==> 6(x^2-6x+8)=0` Factor out the greatest common factor

`6(x-4)(x-2)=0`

** Find p,q such that pq=8 and p+q=-6. Then p=-4 and q=-2 (or vice versa.) Or write `6(x^2-4x-2x+8)=6[x(x-4)-2(x-4)]=6(x-4)(x-2)=0`

Now 6(x-4)(x-2)=0 ==> x=4 or x=2.

x<2       x=2        2<x<4        x=4        x>4
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S'(x)         pos        0               neg          0             pos

Convenient test values are x=0,x=2,x=3,x=4,x=5: when x=0 you have 6(-4)(-2)>0 etc... If `S'(x)>0` the function is increasing.

(2) `S'(x)=6x^2-36x+48`

`S''(x)=12x-36`

S''(x)=0 ==> 12x-36=0 ==> x=3

x<3                x=3              x>3
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S''(x)        neg                  0                 pos