SOLUTION: The curve y=ax^3+2x^2+a^2x+b has a minimum point at (-1,0). Find a and b.

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Question 1044048: The curve y=ax^3+2x^2+a^2x+b has a minimum point at (-1,0). Find a and b.
Answer by josgarithmetic(39615) About Me  (Show Source):
You can put this solution on YOUR website!
dy%2Fdx=3ax%5E2%2B4x%2Ba%5E2=0----Must be for x=-1.

Let x=-1;
3a-4%2Ba%5E2=0, from the derivative being 0 when x is -1;


ALSO from the original equation,
-a%2B2-a%5E2%2Bb=0
so the problem gives the system of equations,


system%28a%5E2%2B3a-4=0%2Ca%5E2%2Ba-2-b=0%29
Solve this system (first, for "a", and then for b).

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First equation of the system is factorable.
%28a-4%29%28a%2B1%29=0
system%28a=-1%2Cor%2Ca=4%29
and the description gave, ",... has a minimum point at (-1,0)".

Either the equation becomes y=-x%5E3%2B2x%5E2%2Bx%2Bb or y=4x%5E3%2B2x%5E2%2B16x%2Bb. Do you believe that the second-derivative might give further information about x at -1 being minimum or maximum?