Proving $f(2) \geq 27$ with Positive Coefficients

In summary, proving $f(2) \geq 27$ with positive coefficients means showing that the value of the function $f(x)$ at the point $x=2$ is greater than or equal to 27, and that all the coefficients in the function are positive. It is important because it allows us to make conclusions about the behavior of the function and ensures a positive trend overall. We can use mathematical techniques such as algebraic manipulation, calculus methods, or graphing to prove this. Positive coefficients are numbers that are greater than zero and are used to multiply variables in a mathematical expression. We cannot prove $f(2) \geq 27$ with negative coefficients as they would result in a negative value for the
  • #1
Albert1
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given :$f(x)=x^3+ax^2+bx+1$ with $a\geq 0 \,\, and \,\, b\geq 0$ and also all of the three roots of $f(x)=0$ are real numbers,

prove $f(2)\geq 27$
 
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  • #2
Albert said:
given :$f(x)=x^3+ax^2+bx+1$ with $a\geq 0 \,\, and \,\, b\geq 0$ and also all of the three roots of $f(x)=0$ are real numbers,

prove $f(2)\geq 27$
using Descartes' Rule of Signs all of the three roots of $f(x)=0---(1)$ must be negative
and let $-x_1,-x_2,-x_3$ be the solutions of $(1)$
(here $x_1,x_2,x_3>0$)
from Vieta's formulas and $AP\geq GP$ we have:
$x_1x_2x_3=1---(2)$
from $(2)$
$a=x_1+x_2+x_3\geq 3\sqrt[3]{x_1x_2x_3}=3---(3)\\
b=x_1x_2+x_1x_3+x_2x_3\geq 3\sqrt[3]{x_1^2x_2^2x_3^2}=3---(4)\\$
from $(3)(4):f(2)\geq 27=2^3+3\times 2^2+3\times 2+1=8+12+6+1$
 
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FAQ: Proving $f(2) \geq 27$ with Positive Coefficients

What does "proving $f(2) \geq 27$ with positive coefficients" mean?

Proving $f(2) \geq 27$ with positive coefficients means that we want to show that the value of the function $f(x)$ at the point $x=2$ is greater than or equal to 27, and that all the coefficients in the function are positive.

Why is proving $f(2) \geq 27$ with positive coefficients important?

Proving $f(2) \geq 27$ with positive coefficients is important because it allows us to make conclusions about the behavior of the function at a specific point, and it also ensures that the function has a positive trend overall. This can be useful in various applications, such as optimization problems.

How can we prove $f(2) \geq 27$ with positive coefficients?

To prove $f(2) \geq 27$ with positive coefficients, we can use mathematical techniques such as algebraic manipulation, calculus methods, or graphing to analyze the behavior of the function and determine if the given inequality holds true.

What are positive coefficients?

Positive coefficients are numbers that are greater than zero and are used to multiply variables in a mathematical expression. In the context of proving $f(2) \geq 27$ with positive coefficients, it means that all the numbers that are multiplied by the variables in the function are positive.

Can we prove $f(2) \geq 27$ with negative coefficients?

No, we cannot prove $f(2) \geq 27$ with negative coefficients. This is because negative coefficients would result in a negative value for the function at the point $x=2$, which contradicts the given inequality. Additionally, proving $f(2) \geq 27$ with negative coefficients would not align with the goal of showing a positive trend for the function.

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