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To solve this problem, we need to analyze the function \( g(t) = -3t^2 + 18t \), which represents the glucose level over time \( t \) after a high-protein meal.

The question is asking for the time when the glucose level is at its maximum, which means finding the critical points by setting the derivative \( g'(t) \) equal to zero.

### Steps:

1. **Differentiate \( g(t) \) with respect to \( t \):**
   \[
   g(t) = -3t^2 + 18t
   \]
   Using the power rule:
   \[
   g'(t) = -6t + 18
   \]

2. **Set \( g'(t) = 0 \) to find the critical points:**
   \[
   -6t + 18 = 0
   \]
   Solving for \( t \):
   \[
   -6t = -18
   \]
   \[
   t = 3
   \]

3. **Verify that this gives a maximum:**
   Since \( g(t) \) is a quadratic function with a negative coefficient for \( t^2 \) (indicating a downward-facing parabola), \( t = 3 \) represents the maximum point.

### Conclusion:
The glucose level reaches its maximum at \( t = 3 \).

Would you like more details on the derivative calculation, or do you have any questions?

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### Related Questions

1. What is the significance of the sign of the coefficient of \( t^2 \) in determining the maximum or minimum?
2. How would the answer change if the equation was \( g(t) = 3t^2 - 18t \)?
3. What is the value of the maximum glucose level at \( t = 3 \)?
4. How would you find the minimum glucose level if the function was different?
5. What is the role of critical points in finding maxima and minima?

#### Tip:
For any quadratic function in the form \( ax^2 + bx + c \), the maximum or minimum occurs at \( t = -\frac{b}{2a} \).

In the medical study, the patient’s blood glucose level (g) versus time after eating a high protein meal followed the pattern g(t) = -3t^2 + 18t. Find the time at which the glucose level was at a maximum. In other words, find when g'(t) = 0.

Learn how to determine the maximum point of the quadratic function g(t) = -3t^2 + 18t, representing blood glucose levels after a meal, by finding when g'(t) = 0. This problem covers differentiation, critical points, and analyzing quadratic functions.

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