Math  /  Algebra

QuestionFind a formula for the family of cubic polynomials with an inflection point at the origin. Cubic polynomials are all of the form f(x)=Ax3+Bx2+Cx+Df(x)=A x^{3}+B x^{2}+C x+D. Use A,B,CA, B, C, and DD for the coefficients which cannot be determined using the given information. f(x)=f(x)= \square

Studdy Solution

STEP 1

1. We are dealing with cubic polynomials of the form f(x)=Ax3+Bx2+Cx+Df(x) = Ax^3 + Bx^2 + Cx + D.
2. The polynomial has an inflection point at the origin (0, 0).
3. A, B, C, and D are coefficients, some of which may be determined by the given information.

STEP 2

1. Understand the conditions for an inflection point
2. Apply the conditions to the general cubic polynomial
3. Determine which coefficients can be found
4. Write the final formula

STEP 3

To have an inflection point at the origin, two conditions must be met:
1. The second derivative of the function at x = 0 must be zero.
2. The third derivative of the function at x = 0 must not be zero.

STEP 4

Let's find the first, second, and third derivatives of our general cubic polynomial:
f(x)=Ax3+Bx2+Cx+Df(x) = Ax^3 + Bx^2 + Cx + D f(x)=3Ax2+2Bx+Cf'(x) = 3Ax^2 + 2Bx + C f(x)=6Ax+2Bf''(x) = 6Ax + 2B f(x)=6Af'''(x) = 6A

STEP 5

Now, let's apply the conditions for an inflection point at the origin (x = 0):
1. f(0)=0f''(0) = 0: 6A(0)+2B=06A(0) + 2B = 0 2B=02B = 0 B=0B = 0
2. f(0)0f'''(0) \neq 0: f(0)=6A0f'''(0) = 6A \neq 0 This means AA must not be zero.

STEP 6

From the previous step, we can determine: - B must be 0 - A can be any non-zero value - C and D can be any value (they don't affect the inflection point at the origin)

STEP 7

We can now write the final formula for the family of cubic polynomials with an inflection point at the origin:
f(x)=Ax3+Cx+Df(x) = Ax^3 + Cx + D
where A ≠ 0, and C and D can be any real numbers.
The formula for the family of cubic polynomials with an inflection point at the origin is:
f(x)=Ax3+Cx+D\boxed{f(x) = Ax^3 + Cx + D}
where A ≠ 0, and C and D can be any real numbers.

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