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I'll address each question one at a time, providing explanations or solving where appropriate. Let’s begin with question 90.

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**90. Prove the formula for the position of the maximum or minimum of a quadratic equation \( y = ax^2 + bx + c \) using calculus.**

To find the maximum or minimum of the function \( y = ax^2 + bx + c \), we’ll use calculus by finding the first derivative and setting it to zero to locate the critical points.

1. **Compute the first derivative**:
   \[
   y' = \frac{d}{dx}(ax^2 + bx + c) = 2ax + b
   \]

2. **Set the derivative to zero to find critical points**:
   \[
   2ax + b = 0
   \]

3. **Solve for \( x \)**:
   \[
   x = -\frac{b}{2a}
   \]

Thus, the critical point, which gives the position of the maximum or minimum, is \( x = -\frac{b}{2a} \), matching the formula from precalculus.

Whether this point is a maximum or minimum depends on the sign of \( a \):
   - If \( a > 0 \), the parabola opens upward, and this point is a minimum.
   - If \( a < 0 \), the parabola opens downward, and this point is a maximum.

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Would you like to continue with further detailed explanations for each question? 

Here are some related questions to deepen your understanding:

1. How does the sign of the coefficient \( a \) in \( y = ax^2 + bx + c \) affect the shape of the parabola?
2. How does the discriminant \( b^2 - 4ac \) impact the roots of the quadratic function \( y = ax^2 + bx + c \)?
3. What happens to the position of the vertex if \( b = 0 \) in the quadratic function?
4. How would the formula for the vertex change if we considered a different form of the quadratic function, such as \( y = a(x - h)^2 + k \)?
5. What are the implications of this calculus-based approach on optimization problems in real-world scenarios?

**Tip:** For quadratic functions, identifying the vertex can quickly help you determine the maximum or minimum values without needing to complete the square.

Prove the formula for the position of the maximum or minimum of a quadratic equation y = ax^2 + bx + c, which was h = -b / (2a), using calculus.

Explore how to derive the vertex formula h = -b / (2a) for quadratic equations using calculus. This step-by-step explanation covers finding critical points by differentiation and is suitable for Grade 12 or Pre-Calculus students.

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