Step 1: Understanding the Concept:
We need to find the infimum (greatest lower bound) and supremum (least upper bound) of a set S defined by an inequality. This involves solving the inequality to determine the interval(s) that define the set.
Step 2: Key Formula or Approach:
1. Solve the inequality \(3x^2 < x^3\).
2. Rewrite the inequality as \(x^3 - 3x^2 > 0\).
3. Factor the polynomial and analyze its sign to find the values of x for which the inequality holds.
4. Identify the infimum and supremum from the resulting set.
Step 3: Detailed Explanation:
We need to solve the inequality:
\[ 3x^2 < x^3 \]
Move all terms to one side:
\[ x^3 - 3x^2 > 0 \]
Factor out the common term \(x^2\):
\[ x^2(x - 3) > 0 \]
For this product to be positive, both factors must have the same sign.
We know that \(x^2\) is always non-negative.
- If \(x=0\), then \(x^2=0\), and the inequality \(0 > 0\) is false. So \(x \neq 0\).
- If \(x \neq 0\), then \(x^2 > 0\).
Since \(x^2\) is positive, for the product \(x^2(x-3)\) to be positive, the other factor \((x-3)\) must also be positive.
\[ x - 3 > 0 \]
\[ x > 3 \]
So, the set S is the open interval \((3, \infty)\).
\[ S = (3, \infty) = \{x \in \mathbb{R} \mid x > 3\} \]
Now we find the infimum and supremum of this set.
- The infimum (greatest lower bound) of the interval \((3, \infty)\) is 3.
- The supremum (least upper bound) of the interval \((3, \infty)\) does not exist, as the set is unbounded above. We say the supremum is \(\infty\).
Note on Discrepancy:
None of the given options match the result (infimum=3, supremum=\(\infty\)). This indicates a severe error in the question. The OCR has transcribed the question as \(3x^2 < x^3\). If the question was intended to be, for example, \(3x < x^2\), then \(x^2-3x > 0 \implies x(x-3)>0\), which gives \(S = (-\infty, 0) \cup (3, \infty)\), which is also unbounded. If the question was \(x^3 < 3x^2\), then \(x^2(x-3)<0\), which gives \(x<3\) and \(x \neq 0\), so \(S = (-\infty, 0) \cup (0, 3)\), which is unbounded below. The question seems irredeemably flawed as stated and transcribed.
Step 4: Final Answer:
As per the inequality \(3x^2 < x^3\), the set is \((3, \infty)\), which has an infimum of 3 and is unbounded above (supremum is \(\infty\)). None of the options are correct.