Proving 3x^2 + x^3 > 0 for x ≥ -1

  • Thread starter silvermane
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In summary, the conversation discusses how to prove the inequality 3x^2 + x^3 greater or equal to 0 for x greater or equal to -1. The attempt at a solution involves simplifying the inequality to 3x^2 greater or equal to 0 and considering different cases. Suggestions are made to expand (1+x)^3 and to factor out an x^2. Ultimately, the conversation concludes with the successful factoring and breaking down of the inequality into cases.
  • #1
silvermane
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Homework Statement


We have x greater or equal to -1, and
(1+x)^3 greater/equal to 1 +3x.
We need to prove that 3x^2 +x^3 is greater or equal to zero for x greater/equal to -1.

The Attempt at a Solution



I've simplified the inequality to 3x^2 greater/equal to 0, but now I need to show that this simplified inequality is true for x greater/equal to -1. I'm almost there, but need some guidance. Thanks for all your help in advance!
 
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  • #2
Just a thought, but perhaps you can do it on a case-by-case basis, the first case being [tex]x > 0[/tex] and the second [tex]-1 \le x < 0[/tex]. But I don't think this inequality is true for all x.

edit: Damn, can you see my LaTeX code or does it just show as an explanation that it is such?
 
Last edited:
  • #3
Ryker said:
edit: Damn, can you see my LaTeX code or does it just show as an explanation that it is such?

I can't see any of your code :(

Edit: I'm trying to prove it true for x greater/equal to -1 though
 
  • #4
Alright, I fixed the code.
 
  • #5
Okay, that works :) thank you!
 
  • #6
Expand (1+x)^3. See where that gets you.
 
  • #7
Alternatively, factor out an x^2.
 
  • #8
Vanadium 50 said:
Alternatively, factor out an x^2.

I've completely factored and broke it into cases. Thanks for the help!
 

Related to Proving 3x^2 + x^3 > 0 for x ≥ -1

What is the statement being proved?

The statement being proved is 3x^2 + x^3 > 0 for x ≥ -1.

Why is this statement important?

This statement is important because it represents a fundamental concept in mathematics known as polynomial inequalities. It is also used in various fields of science, such as physics and engineering, to model and solve real-world problems.

What is the approach for proving this statement?

The approach for proving this statement is by using the properties of inequalities and basic algebraic manipulations to show that the statement holds true for all values of x ≥ -1.

What is the significance of the restriction on the value of x?

The restriction on the value of x (x ≥ -1) is significant because it ensures that the statement is valid for all real numbers. Without this restriction, the statement may not hold true for certain values of x, making the proof incomplete.

Can this statement be proven using different methods?

Yes, there are multiple methods for proving this statement, such as using the quadratic formula, completing the square, or graphing the polynomial function. However, the approach may vary depending on the context and purpose of the proof.

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