What Is the Cubic Polynomial f(x) When 16a + 24b = 9?

In summary, the conversation discusses a cubic polynomial with real roots in the interval (0,1) and how to prove that the function of $f$ satisfies the inequality $16a+24b\le 9$. It also asks to find the corresponding function of $f$ when the equality holds.
  • #1
anemone
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Let the function of $f$ be a cubic polynomial such that $f(x)=x^3-\frac{3}{2}x^2+ax+b=0$, with real roots lie in the interval $(0,\,1)$.

Prove that $16a+24b\le 9$. Find the corresponding function of $f$ when the equality holds.
 
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  • #2
anemone said:
Let the function of $f$ be a cubic polynomial such that $f(x)=x^3-\frac{3}{2}x^2+ax+b=0$, with real roots lie in the interval $(0,\,1)$.

Prove that $16a+24b\le 9$. Find the corresponding function of $f$ when the equality holds.

For ease of notation I will use $A$ as $a$ and $B$ as $b$.

$$(x-a)(x-b)(x-c)=x^3-(a+b+c)x^2+(ab+ac+bc)x-abc$$
$$\Rightarrow A=ab+ac+bc,B=-abc,a+b+c=\dfrac32$$

$$(a+b+c)^2=a^2+b^2+c^2+2A=\dfrac94\Rightarrow a^2+b^2+c^2=\dfrac94-2A\quad(1)$$

$$(a-b)^2\ge0$$

$$a^2+b^2\ge2ab$$

Similarily,

$$a^2+c^2\ge2ac$$

$$b^2+c^2\ge2bc$$

Adding and simplifying gives

$$a^2+b^2+c^2\ge A$$

From $(1)$:

$$\dfrac94-2A\ge A$$

$$\dfrac34\ge A\quad(3)$$

From the AM-GM inequality:

$$\dfrac{a+b+c}{3}\ge\sqrt[3]{abc}\Rightarrow-\dfrac18\le B\quad(4)$$

From $(3)$ and $(4)$

$$12\ge16A$$

$$3\ge-24B$$

$\Rightarrow9\ge16A+24B$ with equality when $a=b=c=\dfrac12$.

The corresponding function for $f$ is $f(x)=\left(x-\dfrac12\right)^3$.
 
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  • #3
greg1313 said:
For ease of notation I will use $A$ as $a$ and $B$ as $b$.

$$(x-a)(x-b)(x-c)=x^3-\dfrac32x^2+(ab+ac+bc)x-abc$$
$$\Rightarrow A=ab+ac+bc,B=-abc$$

$$(a+b+c)^2=a^2+b^2+c^2+2A=\dfrac94\Rightarrow a^2+b^2+c^2=\dfrac94-2A\quad(1)$$

$$(a-b)^2\ge0$$

$$a^2+b^2\ge2ab$$

Similarily,

$$a^2+c^2\ge2ac$$

$$b^2+c^2\ge2bc$$

Adding and simplifying gives

$$a^2+b^2+c^2\ge A$$

From $(1)$:

$$\dfrac94-2A\ge A$$

$$\dfrac34\ge A\quad(3)$$

From the AM-GM inequality:

$$\dfrac{a+b+c}{3}\ge\sqrt[3]{abc}\Rightarrow-\dfrac18\le B\quad(4)$$

From $(3)$ and $(4)$

$$12\ge16A$$

$$3\ge-24B$$

$\Rightarrow9\ge16A+24B$ with equality when $a=b=c=\dfrac12$.

The corresponding function for $f$ is $f(x)=\left(x-\dfrac12\right)^3$.

Awesome, greg1313! And thanks for participating!(Cool)

This problem can still be solved using another route, and I welcome those who are interested to take a stab at it!
 
  • #4
Hint:

Schur's inequality.
 
  • #5
anemone said:
Let the function of $f$ be a cubic polynomial such that $f(x)=x^3-\frac{3}{2}x^2+ax+b=0$, with real roots lie in the interval $(0,\,1)$.

Prove that $16a+24b\le 9$. Find the corresponding function of $f$ when the equality holds.

Solution of other:

Let the three roots be $p,\,q$ and $r$. By Schur's inequality, we have:

$(p+q+r)^3+9pqr\ge 4(p+q+r)(pq+qr+rp)$

which is just

$\left(\dfrac{3}{2}\right)^3+9(-b)\ge 4\left(\dfrac{3}{2}\right)\left(a\right)$

and upon simplification we get:

$16a+24b\le 9$

Equality holds when $p=q=r$, i.e. $f(x)=\left(x-\dfrac{1}{2}\right)^3$.
 
Last edited:

FAQ: What Is the Cubic Polynomial f(x) When 16a + 24b = 9?

What is a cubic equation?

A cubic equation is a polynomial equation of the third degree, meaning it contains terms with the variable raised to the power of three. It has the general form of ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants and a is not equal to 0.

What are the roots of a cubic equation?

The roots of a cubic equation are the values of the variable x that make the equation true. A cubic equation can have up to three distinct roots, and these roots can be real or complex numbers.

How do you find the roots of a cubic equation?

There are several methods for finding the roots of a cubic equation, including factoring, the rational root theorem, and the cubic formula. The most commonly used method is the cubic formula, which involves plugging in the coefficients of the equation into a formula to calculate the roots.

What is the relationship between the roots and the coefficients of a cubic equation?

The relationship between the roots and the coefficients of a cubic equation is given by Vieta's formulas. These formulas state that the sum of the roots is equal to the negative coefficient of x^2 divided by the coefficient of x^3, the product of the roots is equal to the constant term divided by the coefficient of x^3, and the sum of the pairwise products of the roots is equal to the coefficient of x divided by the coefficient of x^3.

Why are cubic equations important?

Cubic equations have many real-world applications in fields such as physics, engineering, and economics. They are used to model and solve problems involving quantities that change over time or space, and are essential for understanding and predicting various phenomena in these fields.

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