Math Problem Statement
Solution
I'll address each question one at a time, providing explanations or solving where appropriate. Let’s begin with question 90.
90. Prove the formula for the position of the maximum or minimum of a quadratic equation using calculus.
To find the maximum or minimum of the function , we’ll use calculus by finding the first derivative and setting it to zero to locate the critical points.
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Compute the first derivative:
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Set the derivative to zero to find critical points:
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Solve for :
Thus, the critical point, which gives the position of the maximum or minimum, is , matching the formula from precalculus.
Whether this point is a maximum or minimum depends on the sign of :
- If , the parabola opens upward, and this point is a minimum.
- If , the parabola opens downward, and this point is a maximum.
Would you like to continue with further detailed explanations for each question?
Here are some related questions to deepen your understanding:
- How does the sign of the coefficient in affect the shape of the parabola?
- How does the discriminant impact the roots of the quadratic function ?
- What happens to the position of the vertex if in the quadratic function?
- How would the formula for the vertex change if we considered a different form of the quadratic function, such as ?
- 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.
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Math Problem Analysis
Mathematical Concepts
Calculus
Quadratic Equations
Differentiation
Formulas
Vertex formula h = -b / (2a)
Derivative of y = ax^2 + bx + c
Theorems
Critical points in calculus
Suitable Grade Level
Grade 12 / Pre-Calculus