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The function y=ax3 + bx2 + cx + d has a maximum turning point at (-2,27) and a minimum turning point at (1,0). Find the values of a b c and d

Please help The function y=ax3 + bx2 + cx + d has a maximum turning point at (-2,27) and-example-1
User Alexwlchan
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Answer: The formula is f(x) = a x ^ 3 + b x ^ 2 + c x + d

f '(x) = 3ax^2 + 2bx + c.

f(- 3) = 3 ==> - 27a + 9b - 3c + d = 3

f '(- 3) = 0 (being a most extreme) ==> 27a - 6b + c = 0.

f(1) = 0 ==> a + b + c + d = 0

f '(1) = 0 (being a base) ==> 3a + 2b + c = 0.

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Along these lines, we have the four conditions

- 27a + 9b - 3c + d = 3

a + b + c + d = 0

27a - 6b + c = 0

3a + 2b + c = 0

Subtracting the last two conditions yields 24a - 8b = 0 ==> b = 3a.

Along these lines, the last condition yields 3a + 6a + c = 0 ==> c = - 9a.

Consequently, we have from the initial two conditions:

- 27a + 9(3a) - 3(- 9a) + d = 3 ==> 27a + d = 3

a + 3a - 9a + d = 0 ==> d = 5a.

Along these lines, a = 3/32 and d = 15/32.

==> b = 9/32 and c = - 27/32.

User Netherwire
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