Question 1163320
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<pre>

Local minimum and local maximum x-values are the roots of the derivative


    f'(x) = 0 = 3ax^2 + 12x + b            (1)


So, you have this system of two equations for two unknowns "a" and "b"


    f'(-1) = 3a*(-1)^2 + 12*(-1) + b = 0   (2)

    f'(2)  = 3a*2^2    + 12*2    + b = 0   (3)


or


     3a + b = 12                            (4)

    12a + b = -24.                          (5)


Subtract equation (4) from equation (5) to get

     9a     = - 36,    a = -36/9 = -4.


Then from equation (4), 

     3*(-4) + b = 12,   b = 12 + 12 = 24.


<U>ANSWER</U>.  a = -4,  b = 24.


Visual check


    {{{graph( 400, 400, -5, 5, -20, 50,
          -4x^3 + 6x^2 + 24x + 4
)}}}


            Plot  y = {{{-4x^3 + 6x^2 + 24x + 4}}}.
</pre>

Done.



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I do not understand, for what need/reason tutor @greenestamps re-told my solution in his post.


It did not become more clear after that.


In any case, thanks for popularizing my solution (!)