Which value is a counterexample to Molly's conjecture?




To find a counterexample to Molly's conjecture that \( x^3 > 3x + 3 \) for all \( x > 0 \), we need to identify a specific value of \( x > 0 \) for which the inequality does not hold. 

Let's start by testing some simple values of \( x \):

1. **For \( x = 1 \):**
   \[
   x^3 = 1^3 = 1
   \]
   \[
   3x + 3 = 3(1) + 3 = 6
   \]
   \[
   \text{Comparison: } 1 < 6
   \]
   The inequality \( x^3 > 3x + 3 \) does **not** hold for \( x = 1 \).

Therefore, \( x = 1 \) is a counterexample to Molly's conjecture. For \( x = 1 \), the inequality \( 1^3 > 3(1) + 3 \) fails because \( 1 \) is not greater than \( 6 \).

Would you like to explore other counterexamples or have any other questions?

### Related Questions:
1. What is the largest value of \( x \) for which \( x^3 < 3x + 3 \) holds?
2. What happens to the inequality when \( x > 2 \)?
3. Can you identify any intervals where the inequality might hold true?
4. How does the inequality behave as \( x \) approaches infinity?
5. Is there a way to modify the conjecture so that it becomes true?

**Tip:** When identifying counterexamples, start with small integer values—they are often easier to calculate and reveal patterns quickly.

Discover how to find a counterexample to Molly's conjecture x^3 > 3x + 3 with a step-by-step solution. Explore simple values of x to identify where the inequality fails. Suitable for high school students studying inequalities in algebra.

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