Minima maxima: Arbitrary constants for a cubic

Submitted by Francis June E… on

Make the curve y=ax³+bx²+cx+d pass through the points (0,1)and (-3,7) and have a critical point at (-1,3)

Ans. y=-x³-4x²-5x+1
Thanks po

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Minima maxima

Jhun Vert Mon, 04/13/2020 - 10:52

Steps:

  1. Substitute (0, 1) to the equation.
    Result: d = 1.
  2. Substitute (-3, 7) to the equation.
    Result: -27a + 9b - 3c = 6.
  3. Substitute (-1, 3) to the equation.
    Result: -a + b - c = 2.
  4. Differentiate and set y' to zero. Because (-1, 3) is a critical point, replace x with -1.
    Result: 3a - 2b + c = 0.
  5. Solve the three equations from steps (2), (3), and (4).

 

Alexander Mon, 04/13/2020 - 12:49

Eq. (2):   -27a + 9b - 3c = 6
Eq. (3):   -a + b - c = 2
Eq. (4):   3a - 2b + c = 0
 

Eq. (2) - 3*Eq. (3)
-24a + 6b = 0
6b = 24a
b = 4a
 

Eq. (3) + Eq. (4)
2a - b = 2
2a - 4a = 2
-2a = 2
a = -1