How to i graph this

Question

H(x)=-3x^4+21x^3-48x^2+36x

 

using a graphing calculator,graaph the function H on the interval o less than or equal to x less than or equal to 3

Answer 1

 

 

zero is a solution since -3x can be factored out

 

the remaining polynomial is

     x^3  -7x^2 + 16x -12

 

By rational root theorem the possible rational zeros are {+-1 ,+-2, +-3, +-4, +-6, +-12}

 2 is a zero since 2^3 - 7(2)^2 + 16(2) - 12 = 8 - 28 + 32 - 12 = -20 +32-12 = 12 - 12 = 0

 

Synthetic division says:

  

 2 |  1     -7     16     -12

               2    -10      12

   _______________________

        1    -5      6         0

 

the quadratic that remains when (x-2) is factored is:

 x^2 - 5x + 6 = ( x   -  3  )(x    -       2  )

 

So the four zeros are 0,2,2,and 3

 

Since x=2 has multiplicity 2, it will "Bounce" at x=2

 

All that remains is to set up a sign table for the following intervals:

 x<0

 0 < x < 2

 2 < x  < 3

 x>3

 

where this function is negative, positive, positive, and negative, respectively.

 

the max occurs around (0.557,8.5) and another local max at (2.693,1.191)