Math Problem Statement

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.

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 y=ax2+bx+cy = ax^2 + bx + c using calculus.

To find the maximum or minimum of the function y=ax2+bx+cy = 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=ddx(ax2+bx+c)=2ax+by' = \frac{d}{dx}(ax^2 + bx + c) = 2ax + b

  2. Set the derivative to zero to find critical points: 2ax+b=02ax + b = 0

  3. Solve for xx: x=b2ax = -\frac{b}{2a}

Thus, the critical point, which gives the position of the maximum or minimum, is x=b2ax = -\frac{b}{2a}, matching the formula from precalculus.

Whether this point is a maximum or minimum depends on the sign of aa:

  • If a>0a > 0, the parabola opens upward, and this point is a minimum.
  • If a<0a < 0, 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:

  1. How does the sign of the coefficient aa in y=ax2+bx+cy = ax^2 + bx + c affect the shape of the parabola?
  2. How does the discriminant b24acb^2 - 4ac impact the roots of the quadratic function y=ax2+bx+cy = ax^2 + bx + c?
  3. What happens to the position of the vertex if b=0b = 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(xh)2+ky = 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.

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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