CALCULUS HELP?? the graph of the function f(x)=x^3+px^2+qx has a point of?

f(x) = x³ + px² + qx

The point (2,-22) is on the graph. This means that f(2) = -22.

f(2) = 8 + 4p + 2q = -22 ==> 4p+2q = -30....(1)

Since this is an inflection point, f '' (2) = 0.

f ' (x) = 3x² + 2px

f ''(x) = 6x + 2p

f '' (2) = 12+2p = 0 ==> 2p = -12 ==> p = -6

Plug p=-6 into (1):

-24 + 2q = -30 ==> 2q =-6 ==> q = -3

derivative equations f(x) = x^3 + px^2 + qx

f'(x) = 3x^2 +2px

f''(x) = 6x + 2p

when f''(x) = 0

6x + 2p = 0

<=> 6x = - 2p

<=> x = - p/3

I has a point of inflection at (2,-22) => x = -p/3 = 2

=> p = - 6

Replace p =- 6 in f(x) = x^3 + px^2 + qx

f(x) = x^3 - 6x^2 + qx (*)

at this replace A (2 ; -22) in (*) with y = f(2) = -22

-22 = 8 - 6.4 + 2q

-22 = 2q - 16

2q = -6

=> q = - 3

So p = -6 , q = -3 the graph of the function f(x)=x^3+px^2+qx has a point of inflection at (2,-22)

x-axis value of inflection points are root of the 2nd derivative then:

f ' ' (x) = 6x +2p = 0 then

f ' ' (2) = 6*2+2p =0 then p = -6

Now f(x) = x^3 - 6x^2 +qx and f(2) = -22

8 - 24 +2q = 22

q = 19

f (x) = x^3 - 6x^2 + 19x

First change ( x=2, y= -22 ) to get the first eq 8 +4p +2q = - 22 ( eq a million) inflection aspect exhibits the 2d spinoff f ' ' ( x) = 6 x +2p = 0 and 0 = 12 + 2p ( eq 2 ) remedy equipment (a million) and (2) ==> ( p= -6 ,q = - 3 )