To solve this, we use the Squeeze Theorem, which states that if a function $f(x)$ is squeezed between two functions and they both converge to the same limit as $x$ approaches a point, then $f(x)$ must also converge to that same limit.

The given inequalities are: $-3x^2 + 12x + 81 \leq f(x) \leq 3x^2 - 12x + 105 \quad \text{for all} \quad 0 \leq x \leq 3$

Step 1: Evaluate the lower bound as $x \to 0^+$:

$\lim_{x \to 0^+} \left( -3x^2 + 12x + 81 \right) = -3(0)^2 + 12(0) + 81 = 81$

Step 2: Evaluate the upper bound as $x \to 0^+$:

$\lim_{x \to 0^+} \left( 3x^2 - 12x + 105 \right) = 3(0)^2 - 12(0) + 105 = 105$

Step 3: Apply the Squeeze Theorem

As $x \to 0^+$, we have: $81 \leq f(x) \leq 105$ Since the lower and upper bounds give different limits (81 and 105), the function $f(x)$ is bounded but does not converge to a single value as $x \to 0^+$.

Conclusion:

There is insufficient information to determine its exact value because the function is squeezed between two values that do not converge to the same limit.

So, the correct statement is:

• There is insufficient information to determine its exact value.

Would you like more details, or do you have any questions? Here are 5 questions related to this topic:

1. How does the Squeeze Theorem work in calculus?
2. What happens if the bounds converge to the same value?
3. What other methods can be used to evaluate limits?
4. Can we always use the Squeeze Theorem for inequalities?
5. How can discontinuities affect limits?

Tip: The Squeeze Theorem is particularly useful when dealing with complex or oscillatory functions where direct evaluation is difficult.