help with some maths question please?

[匿名]

2012-01-26 6:13 am

hi people, I'm doing revision for maths and there are a few questions here that I can't solve.. can you guys help me please? So here are the questions:

Question 1: A cubic polynomial with a critical point at x=2, an inflection point at (1,4), and a leading coefficient of 1.

oh sorry i think that's all haha ^-^ thanks for all of your help!

回答 (1)

Roger the Mole

2012-01-26 9:59 pm

✔ 最佳答案

The model for such a cubic would be:
x^3 + Ax^2 + Bx + C = y
Then the problem becomes, solve for A, B and C.
Take the derivative and set it to zero to find the critical and inflection points:
3x^2 + 2Ax + B = 0
Substitute the two given values for x, one at a time:
3(2)^2 + 2A(2) + B = 0
3(1)^2 + 2A(1) + B = 0
Expand and combine like terms:
4A + B = -12
2A + B = -3
Solve simultaneously:
A = -9/2
B = 6
Now the model equation becomes:
x^3 - (9/2)x^2 + 6x + C = y
Substitute the values for x and y from the given point (1/4):
(1)^3 - (9/2)(1)^2 + 6(1) + C = 4
Expand and combine like terms:
C = 3/2
So the desired equation is:
x^3 - (9/2)x^2 + 6x + (3/2) = y

If this was supposed to be done without calculus I'm stumped.

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