Sketch the graph of the given polynomial function,and show work

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f(x) = x^3 - 3x^2 - 9x + 6        x values: -3,-2,-1,0,1,2,3,4

Oct 25th, 2015

Thank you for the opportunity to help you with your question!

We have the polynomial function as y=x^3-3x^2-9x+6

To graph the function,we have to find the point of maxima and minima.

Differentiate the function with respect to x and equate it to zero,

d[f(x)]/dx=d(x^3-3x^2-9x+6)/dx=3x^2-6x-9 =0


Solve for x




x=-1 and x=3

To determine whether maxima or minima occurs differentiate again and substitute the critical value of x

if d^2y/dx^2 >0 then minima occurs

and d^2/dx^2 <0 ,maxima occurs

Now,d^2y/dx^2 =d(f'(x)/dx=d(x^2-2x-3)/dx=2x-2

At x=-1 d^2y/dx^2 <0 therefore maxima occurs and maximum value is f(-1)=11

at x=3 minima ocur as d^2y/dsx^2 >0 ,At x=3 f(3)=-21

Also the y intercept are (0,6)

Using the above information creating a table of values

x             y

-3            -21

-2            4

-1            11(maxima)

0                     6

1                  -5

2                     -16

3              -21(minima)

Please let me know if you need any clarification. I'm always happy to answer your questions.
Oct 25th, 2015

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