Critical Numbers and Intervals of Decrease

Oct 13, 2025 by windailys.com 49 views

Critical Numbers and Intervals of Decrease

Understanding Critical Numbers

Critical numbers are values of x that make the derivative of a function equal to zero or undefined. These points are crucial in determining the behavior of a function, including its local maxima and minima. In this article, we will explore the concept of critical numbers and intervals of decrease for a given function.

The Function P(x) = 8x√(1-x^2)

The function P(x) = 8x√(1-x^2) is a product of two functions: 8x and √(1-x^2). To find the critical numbers, we need to find the derivative of P(x) and set it equal to zero or undefined.

Finding the Derivative

To find the derivative of P(x), we can use the product rule and the chain rule.

P'(x) = d(8x)/dx * √(1-x^2) + 8x * d(√(1-x^2))/dx

P'(x) = 8 * √(1-x^2) + 8x * (1/2) * (1-x²⁾(-1/2) * (-2x)

P'(x) = 8 * √(1-x^2) - 4x^2 / √(1-x^2)

Simplifying the Derivative

We can simplify the derivative by combining the two terms.

P'(x) = (8 * √(1-x^2) - 4x^2) / √(1-x^2)

Finding Critical Numbers

To find the critical numbers, we need to set the derivative equal to zero or undefined.

(8 * √(1-x^2) - 4x^2) / √(1-x^2) = 0

We can multiply both sides by √(1-x^2) to eliminate the denominator.

8 * √(1-x^2) - 4x^2 = 0

We can rearrange the equation to isolate the square root term.

8 * √(1-x^2) = 4x^2

Dividing both sides by 8, we get:

√(1-x^2) = (1/2) * x^2

Squaring both sides, we get:

1 - x^2 = (1/4) * x^4

Rearranging the equation, we get:

(1/4) * x^4 + x^2 - 1 = 0

This is a quadratic equation in x^2. We can solve it using the quadratic formula.

x^2 = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1/4, b = 1, and c = -1.

x^2 = (-(1) ± √((1)^2 - 4(1/4)(-1))) / (2(1/4))

x^2 = (-1 ± √(1 + 1)) / (1/2)

x^2 = (-1 ± √2) / (1/2)

x^2 = (-1 ± √2) * 2

x^2 = -2 ± 2√2

x = ±√(-2 ± 2√2)

Critical Numbers

The critical numbers are the values of x that make the derivative equal to zero or undefined. In this case, the critical numbers are:

x = ±√(-2 + 2√2) x = ±√(-2 - 2√2)

The Function f(x) = x^3 - 9x + 6

The function f(x) = x^3 - 9x + 6 is a cubic function. To find the intervals on which the function is decreasing, we need to find the derivative of f(x) and determine its sign.

Finding the Derivative

To find the derivative of f(x), we can use the power rule.

f'(x) = d(x^3)/dx - d(9x)/dx + d(6)/dx

f'(x) = 3x^2 - 9

Finding Intervals of Decrease

To find the intervals on which the function is decreasing, we need to determine the sign of the derivative.

f'(x) = 3x^2 - 9

We can set the derivative equal to zero to find the critical points.

3x^2 - 9 = 0

x^2 = 3

x = ±√3

Intervals of Decrease

The function is decreasing on the intervals where the derivative is negative. In this case, the derivative is negative when:

x < -√3 or x > √3

Conclusion

In this article, we explored the concept of critical numbers and intervals of decrease for two given functions. We found the critical numbers for the function P(x) = 8x√(1-x^2) and determined the intervals on which the function f(x) = x^3 - 9x + 6 is decreasing. The critical numbers are the values of x that make the derivative equal to zero or undefined, and the intervals of decrease are the intervals where the derivative is negative.
Q&A: Critical Numbers and Intervals of Decrease

Q: What are critical numbers?

A: Critical numbers are values of x that make the derivative of a function equal to zero or undefined. These points are crucial in determining the behavior of a function, including its local maxima and minima.

Q: How do I find the critical numbers of a function?

A: To find the critical numbers, you need to find the derivative of the function and set it equal to zero or undefined. Then, solve for x to find the critical numbers.

Q: What is the difference between a critical number and a local maximum or minimum?

A: A critical number is a value of x that makes the derivative equal to zero or undefined. A local maximum or minimum is a point on the graph of the function where the function changes from increasing to decreasing or vice versa.

Q: How do I determine the intervals on which a function is increasing or decreasing?

A: To determine the intervals on which a function is increasing or decreasing, you need to find the derivative of the function and determine its sign. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing.

Q: What is the significance of the derivative in determining the behavior of a function?

A: The derivative of a function represents the rate of change of the function with respect to x. It is used to determine the behavior of the function, including its local maxima and minima, and the intervals on which the function is increasing or decreasing.

Q: Can a function have multiple critical numbers?

A: Yes, a function can have multiple critical numbers. In fact, most functions have multiple critical numbers.

Q: How do I determine the type of critical number (local maximum or minimum) a point is?

A: To determine the type of critical number a point is, you need to examine the behavior of the function on either side of the critical point. If the function changes from increasing to decreasing at the critical point, it is a local maximum. If the function changes from decreasing to increasing at the critical point, it is a local minimum.

Q: Can a function have a critical number that is not a local maximum or minimum?

A: Yes, a function can have a critical number that is not a local maximum or minimum. This can occur when the function has a point of inflection, where the function changes from concave up to concave down or vice versa.

Q: How do I find the intervals on which a function is concave up or concave down?

A: To find the intervals on which a function is concave up or concave down, you need to find the second derivative of the function and determine its sign. If the second derivative is positive, the function is concave up. If the second derivative is negative, the function is concave down.

Q: What is the significance of the second derivative in determining the behavior of a function?

A: The second derivative of a function represents the rate of change of the first derivative with respect to x. It is used to determine the concavity of the function and the points of inflection.

Q: Can a function have multiple points of inflection?

A: Yes, a function can have multiple points of inflection. In fact, most functions have multiple points of inflection.

Q: How do I determine the type of point of inflection (concave up or concave down) a point is?

A: To determine the type of point of inflection a point is, you need to examine the behavior of the function on either side of the point of inflection. If the function changes from concave up to concave down at the point of inflection, it is a point of inflection where the function changes from concave up to concave down. If the function changes from concave down to concave up at the point of inflection, it is a point of inflection where the function changes from concave down to concave up.