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 $f(x)=A x^{3}+B x^{2}+C x+D$ . Use $A, B, C$ , and $D$ for the coefficients which cannot be determined using the given information. $f(x)=$ $\square$

Studdy Solution

STEP 1

1. We are given a general cubic polynomial $f(x) = Ax^3 + Bx^2 + Cx + D$ .
2. The polynomial has an inflection point at the origin, which is the point $(0,0)$ .
3. An inflection point is where the second derivative changes sign, meaning the second derivative at that point is zero.

STEP 2

1. Determine the conditions for an inflection point at the origin.
2. Calculate the first and second derivatives of the cubic polynomial.
3. Apply the condition for the inflection point to the second derivative.
4. Simplify the polynomial using the condition derived from the inflection point.
5. Write the final form of the polynomial.

STEP 3

To have an inflection point at the origin, the second derivative of the polynomial must be zero at $x = 0$ .

STEP 4

Calculate the first derivative of $f(x)$ :
$f'(x) = 3Ax^2 + 2Bx + C$
Calculate the second derivative of $f(x)$ :
$f''(x) = 6Ax + 2B$

STEP 5

Apply the condition for the inflection point:
Set the second derivative equal to zero at $x = 0$ :
$f''(0) = 6A(0) + 2B = 0$ $2B = 0$ $B = 0$

STEP 6

Substitute $B = 0$ into the original polynomial:
$f(x) = Ax^3 + Cx + D$

STEP 7

The final form of the polynomial with an inflection point at the origin is:
$f(x) = Ax^3 + Cx + D$
The formula for the family of cubic polynomials with an inflection point at the origin is:
$\boxed{Ax^3 + Cx + D}$

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

Studdy Solution

STEP 1

1. We are given a general cubic polynomial $f(x) = Ax^3 + Bx^2 + Cx + D$ .
2. The polynomial has an inflection point at the origin, which is the point $(0,0)$ .
3. An inflection point is where the second derivative changes sign, meaning the second derivative at that point is zero.

STEP 2

STEP 3

To have an inflection point at the origin, the second derivative of the polynomial must be zero at $x = 0$ .

STEP 4

Calculate the first derivative of $f(x)$ :
$f'(x) = 3Ax^2 + 2Bx + C$
Calculate the second derivative of $f(x)$ :
$f''(x) = 6Ax + 2B$

STEP 5

Apply the condition for the inflection point:
Set the second derivative equal to zero at $x = 0$ :
$f''(0) = 6A(0) + 2B = 0$ $2B = 0$ $B = 0$

STEP 6

Substitute $B = 0$ into the original polynomial:
$f(x) = Ax^3 + Cx + D$

STEP 7

The final form of the polynomial with an inflection point at the origin is:
$f(x) = Ax^3 + Cx + D$
The formula for the family of cubic polynomials with an inflection point at the origin is:
$\boxed{Ax^3 + Cx + D}$

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Studdy Solution

STEP 1

STEP 2

STEP 3

To have an inflection point at the origin, the second derivative of the polynomial must be zero at x=0 x = 0 x=0.

STEP 4

Calculate the first derivative of f(x) f(x) f(x):f′(x)=3Ax2+2Bx+C f'(x) = 3Ax^2 + 2Bx + C f′(x)=3Ax2+2Bx+CCalculate the second derivative of f(x) f(x) f(x):f′′(x)=6Ax+2B f''(x) = 6Ax + 2B f′′(x)=6Ax+2B

STEP 5

Apply the condition for the inflection point:Set the second derivative equal to zero at x=0 x = 0 x=0:f′′(0)=6A(0)+2B=0 f''(0) = 6A(0) + 2B = 0 f′′(0)=6A(0)+2B=0 2B=0 2B = 0 2B=0 B=0 B = 0 B=0

STEP 6

Substitute B=0 B = 0 B=0 into the original polynomial:f(x)=Ax3+Cx+D f(x) = Ax^3 + Cx + D f(x)=Ax3+Cx+D

STEP 7

The final form of the polynomial with an inflection point at the origin is:f(x)=Ax3+Cx+D f(x) = Ax^3 + Cx + D f(x)=Ax3+Cx+D The formula for the family of cubic polynomials with an inflection point at the origin is:Ax3+Cx+D \boxed{Ax^3 + Cx + D} Ax3+Cx+D​

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To have an inflection point at the origin, the second derivative of the polynomial must be zero at $x = 0$ .

Calculate the first derivative of $f(x)$ :
$f'(x) = 3Ax^2 + 2Bx + C$
Calculate the second derivative of $f(x)$ :
$f''(x) = 6Ax + 2B$

Apply the condition for the inflection point:
Set the second derivative equal to zero at $x = 0$ :
$f''(0) = 6A(0) + 2B = 0$ $2B = 0$ $B = 0$

Substitute $B = 0$ into the original polynomial:
$f(x) = Ax^3 + Cx + D$

The final form of the polynomial with an inflection point at the origin is:
$f(x) = Ax^3 + Cx + D$
The formula for the family of cubic polynomials with an inflection point at the origin is:
$\boxed{Ax^3 + Cx + D}$