SOLUTION: The function {{{y=ax^3+bx^2+cx+d}}} has a relative maximum at the point (-2,27) and a relative minimum at the point (1,0). Find the values of a, b, c and d. I end up with 4 equa

Algebra.Com
Question 1151382: The function has a relative maximum at the point (-2,27) and a relative minimum at the point (1,0). Find the values of a, b, c and d.
I end up with 4 equations the I need to solve simultaneously. Is there a better method?
Answer by ikleyn(52921)   (Show Source): You can put this solution on YOUR website!
.

First derivative is

    y' = 3ax^2 + 2bx + c.


According to the condition, it has zeroes at x= -2 and x= 1.


Hence,   = -  = 1  and   = -2   

(according to Vieta's theorem).



From these equalities,

    b =   and  c = -6a.                (*)



Substitute these values to the equation for y.  You will get

    y =  +  -  + d.         (1)



Substitute x = -2  into (1)  to get y = 27.  It gives you

    27 = -8a + 6a + 12a + d,      or

    27 = 10a + d.                            (2)



Substitute x = 1  into (1)  to get y = 0.  It gives you

    0 = a +  - 6a  + d,      or

    0 = 2a + 3a - 12a + 2d,       or

    0 = -7a + 2d.                            (3)


Thus you have the system of 2 equations in 2 unknowns


    10a + d = 27,    (2')

    -7a + 2d = 0.    (3')


From (2'), express d = 27-10a  and substitute it into (3').  You will get

    -7a + 2*(27-10a) = 0,

    -7a + 54 - 20a   = 0

     54              = 7a + 20a

     54              = 27a

      a              = 54/27 = 2.


Thus  a = 2,  d = 27 - 10a = 27 - 20 = 7.


Finally,  from (*)  

     b =  =  = 3,   and 

     c = -6a = -6*2 = -12.


ANSWER.  a = 2;  b = 3;  c = -12;  d = 7.


Solved.



RELATED QUESTIONS

The function y = ax^3 + bx^2 + cx + d has a maximum turning point at (−2, 27) and a... (answered by Solver92311)
find values for a,b,c,and d so that the function f (x) =ax³ + bx² + cx + d has a... (answered by Boreal)
which is true for the graph of f(x)=-x cubed + 3x-2? A)relative maximum at (1,0)... (answered by stanbon)
Determine the values of the constants a and b so that the function f ( x )= x3 + ax2 + bx (answered by greenestamps)
find the relative maximum and relative minimum values of the function below.... (answered by RAY100)
use the graph of the giveb function to find any relative maxima and relative minima.... (answered by stanbon)
Hi can you plz help me with this qs? I didn't know under which section to post it so im... (answered by greenestamps)
A quadratic graph has a maximum point (2,5). Find the equation of the graph in form... (answered by josgarithmetic)
a quadratic graph has maximum point (2,5) and passes through point (0,-3). Find the... (answered by Fombitz)