Given that x is a number greater than 0, Molly conjectured that x3>3x+3.

Molly's conjecture is that for any number \( x > 0 \), the inequality \( x^3 > 3x + 3 \) holds. Let's analyze whether this conjecture is true for all positive \( x \).

### Step 1: Set up the inequality
\[ x^3 > 3x + 3 \]

### Step 2: Subtract \( 3x + 3 \) from both sides to bring everything to one side:
\[ x^3 - 3x - 3 > 0 \]

### Step 3: Analyze the function
Define the function \( f(x) = x^3 - 3x - 3 \). We want to find out where \( f(x) > 0 \).

### Step 4: Determine the critical points
To determine the behavior of \( f(x) \), we first find its derivative and solve for \( f'(x) = 0 \):

\[ f'(x) = 3x^2 - 3 \]

Set \( f'(x) = 0 \):

\[ 3x^2 - 3 = 0 \]
\[ x^2 = 1 \]
\[ x = \pm 1 \]

Since we're only interested in \( x > 0 \), we consider \( x = 1 \).

### Step 5: Analyze the sign of \( f(x) \)
Let's check the sign of \( f(x) \) in intervals around the critical point \( x = 1 \).

1. For \( 0 < x < 1 \), pick a test point, say \( x = 0.5 \):
   \[ f(0.5) = (0.5)^3 - 3(0.5) - 3 = 0.125 - 1.5 - 3 = -4.375 \]
   \( f(x) < 0 \) in this interval.

2. For \( x = 1 \):
   \[ f(1) = 1^3 - 3(1) - 3 = 1 - 3 - 3 = -5 \]
   \( f(1) < 0 \).

3. For \( x > 1 \), pick a test point, say \( x = 2 \):
   \[ f(2) = 2^3 - 3(2) - 3 = 8 - 6 - 3 = -1 \]
   \( f(x) \) starts to approach zero as \( x \) increases but remains negative.

### Conclusion:
From this analysis, we see that \( f(x) = x^3 - 3x - 3 \) is negative for \( x > 0 \) and doesn't become positive as \( x \) increases. Therefore, Molly's conjecture that \( x^3 > 3x + 3 \) for all \( x > 0 \) is **false**.

Would you like to explore this further, or do you have any other questions?

### Related Questions:
1. What is the minimum value of the function \( f(x) = x^3 - 3x - 3 \) for \( x > 0 \)?
2. How does \( x^3 \) compare to other linear functions like \( 2x + 1 \)?
3. Can you find a specific value of \( x > 0 \) where \( x^3 = 3x + 3 \)?
4. What happens to the inequality \( x^3 > 3x + 3 \) if \( x < 0 \)?
5. How can we generalize the inequality \( x^3 > ax + b \) for other constants \( a \) and \( b \)?

**Tip:** When dealing with inequalities involving polynomials, finding the critical points and testing intervals is a powerful method for determining where the inequality holds.

Explore Molly's conjecture that for any number x greater than 0, the inequality x^3 > 3x + 3 holds. Understand the analysis involving polynomial inequalities and critical points. Suitable for advanced high school students (Grades 10-12).

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