Math Problem Statement

4x−x2

Solution

Given the quadratic function y=4xx2y = 4x - x^2, we can find the position of the maximum or minimum using calculus, following a similar procedure as before.

Step 1: Express the quadratic function

The quadratic function is:

y=x2+4xy = -x^2 + 4x

This is in the standard form y=ax2+bx+cy = ax^2 + bx + c, where:

  • a=1a = -1
  • b=4b = 4
  • c=0c = 0

Step 2: Find the first derivative

To find the critical points, take the first derivative of yy with respect to xx:

dydx=ddx(x2+4x)\frac{dy}{dx} = \frac{d}{dx}(-x^2 + 4x)

Applying the power rule:

dydx=2x+4\frac{dy}{dx} = -2x + 4

Step 3: Set the first derivative to zero

Set the first derivative equal to zero and solve for xx:

2x+4=0-2x + 4 = 0

2x=42x = 4

x=2x = 2

Step 4: Identify the nature of the critical point

To determine whether this critical point is a maximum or minimum, find the second derivative:

d2ydx2=ddx(2x+4)=2\frac{d^2y}{dx^2} = \frac{d}{dx}(-2x + 4) = -2

Since the second derivative is negative (2<0-2 < 0), the function is concave down, indicating that the critical point x=2x = 2 is a maximum.

Step 5: Find the corresponding yy-coordinate

Substitute x=2x = 2 back into the original function to find the maximum value:

y=x2+4x=(2)2+4(2)y = -x^2 + 4x = -(2)^2 + 4(2)

y=4+8=4y = -4 + 8 = 4

Conclusion

The maximum value of the function y=4xx2y = 4x - x^2 occurs at x=2x = 2, and the maximum value is y=4y = 4.


Would you like further details or have any questions?

Here are some related questions you might find interesting:

  1. How would you graph the function y=4xx2y = 4x - x^2?
  2. What is the significance of the vertex in the context of this problem?
  3. How does the quadratic function relate to physical problems like projectile motion?
  4. What are the intercepts of the function y=4xx2y = 4x - x^2?
  5. Can you express the function y=4xx2y = 4x - x^2 in vertex form?
  6. What happens to the maximum value if the coefficient of x2x^2 changes?
  7. How would the graph change if the function were y=x24xy = x^2 - 4x?
  8. Can the function y=4xx2y = 4x - x^2 have any other critical points?

Tip: When dealing with quadratic functions, the vertex is always a critical point, and it gives the maximum or minimum value depending on the sign of the leading coefficient.

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Math Problem Analysis

Mathematical Concepts

Quadratic Equations
Calculus
Critical Points
Maximum and Minimum

Formulas

Quadratic function standard form: y = ax^2 + bx + c
First derivative for critical points
Second derivative test for concavity

Theorems

Second Derivative Test

Suitable Grade Level

Grades 11-12