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.