# Inequality Challenge: Prove $x$ for $x>0$

• MHB
• anemone
In summary, the "Inequality Challenge" is a mathematical problem that requires you to prove a statement about inequality, specifically a statement about a variable x greater than 0. To prove this statement, you can use mathematical logic and properties such as algebraic manipulation and inequalities. The significance of x being greater than 0 is that it represents a positive number and the statement being proved may not apply to negative numbers. You can use any mathematically sound method to prove the statement, but some methods may be more efficient or easier to understand. There are also various resources and tools available to assist with the "Inequality Challenge," such as textbooks, online tutorials, math tutors, and online programs.
anemone
Gold Member
MHB
POTW Director
Prove $x+x^9+x^{25}<1+x^4+x^{16}+x^{36}$ for $x>0$.

Let's see if my head is fully back in the game.

Let's look at $$\displaystyle f(x) = x^{36} - x^{25} + x^{16} - x^9 + x^4 - x + 1$$

For x = 0, f(0) = 1.

For 0 < x < 1:
$$\displaystyle x^{36} - (x^{25} - x^{16} ) - (x^9 - x^4) - (x - 1) > 0$$ because each term inside the parentheses are negative.

For 1 < x
$$\displaystyle (x^{36} - x^{25}) + (x^{16} - x^9) + (x^4 - x) + 1 > 0$$ because every term inside the parentheses are positive.

Therefore f(x) has no real zeros on 0 < x.

$$\displaystyle x^{36} - x^{25} + x^{16} - x^9 + x^4 - x + 1 > 0 \implies x^{36} + x^{16} + x^4 + 1 > x^{25} + x^9 + x$$ on 0 < x.

-Dan

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## 1. What is the "Inequality Challenge"?

The "Inequality Challenge" is a mathematical problem that requires proving a statement for a specific range of values. In this case, the statement is about a variable x being greater than 0.

## 2. How is the "Inequality Challenge" relevant to science?

The "Inequality Challenge" is relevant to science because it involves using mathematical reasoning and logic to prove a statement. This is a fundamental aspect of the scientific method and is used to test hypotheses and theories.

## 3. What does it mean to prove something for a specific range of values?

Proving something for a specific range of values means that the statement being proven is only true for certain values of the variable. In this case, the statement is only true for values of x that are greater than 0.

## 4. How can one prove the statement for x>0?

To prove the statement for x>0, one can use various mathematical techniques such as algebra, calculus, or logical reasoning. The specific method used will depend on the complexity of the statement and the available information.

## 5. What is the significance of proving the statement for x>0?

Proving the statement for x>0 is significant because it provides evidence that the statement is true for a specific range of values. This can help to support theories and make predictions in various scientific fields, such as economics, physics, and biology.

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