How To Find Critical Numbers Subject To

Finding critical numbers is a cornerstone of calculus, acting as essential stepping stones for analyzing functions, locating local maxima and minima, and solving optimization problems. This article serves as an in-depth guide to understanding critical numbers, their significance, and the methods to determine them, empowering you to analyze functions effectively.

What Are Critical Numbers?

A critical number, also known as a critical point, of a function f(x) is a value c in the domain of f where either the derivative of f at c is zero (f'(c) = 0) or the derivative of f at c is undefined (f'(c) does not exist). These points are "critical" because they represent potential locations where the function's behavior changes dramatically – from increasing to decreasing (local maximum) or decreasing to increasing (local minimum). They also can indicate endpoints of the domain where the behavior needs examination.

Critical numbers are x-values. The point (c, f(c)) on the graph of the function is called a critical point.

Why Are Critical Numbers Important?

Critical numbers are indispensable tools in calculus for several key reasons:

• Finding Local Extrema: They are fundamental in identifying local maxima and minima of a function. By analyzing the sign of the derivative around critical numbers, we can determine whether a function reaches a local maximum or minimum at that point. This is a core concept in optimization problems.

• Determining Intervals of Increase and Decrease: Critical numbers delineate intervals where a function is either increasing or decreasing. By testing values within these intervals, one can discern the function's behavior across its domain.

• Solving Optimization Problems: Many real-world problems involve maximizing or minimizing a certain quantity (e.g., profit, cost, area). Critical numbers provide a way to find the values of the variables that lead to the optimal solution.

• Graphing Functions Accurately: Knowing the critical numbers and the function's behavior around them greatly aids in sketching an accurate graph of the function. This includes locating turning points and understanding the overall shape.

Step-by-Step Guide to Finding Critical Numbers

Finding critical numbers involves a systematic approach. Here's a detailed breakdown of the steps:

1. Find the Derivative of the Function f'(x)

The first step is to find the derivative of the function f(x). This is done using the standard rules of differentiation:

• Power Rule: If f(x) = xⁿ, then f'(x) = nx^n-1.
• Constant Multiple Rule: If f(x) = cf(x), where c is a constant, then f'(x) = cf'(x).
• Sum/Difference Rule: If f(x) = u(x) ± v(x), then f'(x) = u'(x) ± v'(x).
• Product Rule: If f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x).
• Quotient Rule: If f(x) = u(x) / v(x), then f'(x) = [v(x)u'(x) - u(x)v'(x)] / [v(x)]².
• Chain Rule: If f(x) = u(v(x)), then f'(x) = u'(v(x))v'(x).
• Derivatives of Trigonometric Functions: These are standard derivatives such as d/dx (sin x) = cos x, d/dx (cos x) = -sin x, etc.
• Derivatives of Exponential and Logarithmic Functions: These are standard derivatives such as d/dx (e^x) = e^x, d/dx (ln x) = 1/x, etc.

Example: Let f(x) = 3x⁴ - 8x³ + 6x² - 2. Using the power rule and constant multiple rule, we find the derivative: f'(x) = 12x³ - 24x² + 12x.

2. Find Where f'(x) = 0

Set the derivative equal to zero and solve for x. These x-values are potential critical numbers. f'(x) = 0 means we are looking for points where the tangent line to the graph of f(x) is horizontal.

Example (Continuing from above): Set f'(x) = 12x³ - 24x² + 12x = 0. Factor out 12x: 12x(x² - 2x + 1) = 0. Factor the quadratic: 12x(x - 1)² = 0. Solve for x: x = 0 or x = 1.

3. Find Where f'(x) is Undefined

Determine the values of x for which the derivative f'(x) does not exist. This typically occurs when there is a division by zero, a square root of a negative number (in the real number system), or a similar situation that makes the derivative undefined. These x-values are also potential critical numbers.

Example: Consider the function f(x) = x^2/3. The derivative is f'(x) = (2/3)x^-1/3 = 2 / (3x^1/3). The derivative is undefined when the denominator is zero: 3x^1/3 = 0, which implies x = 0.

4. Check if the Critical Numbers are in the Domain of the Original Function f(x)

It is crucial to ensure that the potential critical numbers found in steps 2 and 3 are actually in the domain of the original function f(x). If a value of x makes the original function undefined, then it cannot be a critical number.

Example: Suppose we found x = -1 as a potential critical number for the function f(x) = √(x + 2). The domain of f(x) is x ≥ -2. Since -1 is in the domain, it is a valid critical number. However, if we found x = -3, this would not be a critical number because it's not in the domain.

5. List All Critical Numbers

Combine the x-values found in steps 2 and 3 that are also within the domain of the original function. These are the critical numbers of the function f(x).

Example (Combining previous examples): For f(x) = 3x⁴ - 8x³ + 6x² - 2, we found x = 0 and x = 1. Both are in the domain (all real numbers), so the critical numbers are x = 0 and x = 1. For f(x) = x^2/3, we found x = 0. This is in the domain (all real numbers), so the critical number is x = 0.

Finding Critical Numbers on a Closed Interval

When finding critical numbers on a closed interval [a, b], the process is the same as above, but with an additional step:

• Find the Critical Numbers within the Interval: Determine the critical numbers of the function using the steps outlined above. However, only include the critical numbers that lie within the open interval (a, b). The endpoints a and b are also important and need to be considered separately.
• Evaluate the Function at the Endpoints and Critical Numbers: Evaluate the function f(x) at the endpoints a and b, and at all critical numbers found within the interval (a, b).
• Identify the Absolute Maximum and Minimum: The largest value of f(x) obtained in the previous step is the absolute maximum of the function on the interval [a, b], and the smallest value is the absolute minimum.

Example: Find the absolute maximum and minimum values of the function f(x) = x³ - 3x² + 1 on the interval [-1/2, 4].

1. Find the Derivative: f'(x) = 3x² - 6x.
2. Find Where f'(x) = 0: 3x² - 6x = 0 => 3x(x - 2) = 0 => x = 0 or x = 2.
3. Find Where f'(x) is Undefined: The derivative is a polynomial, so it is defined everywhere.
4. Critical Numbers within the Interval: Both x = 0 and x = 2 lie within the interval (-1/2, 4).
5. Evaluate the Function:
   • f(-1/2) = (-1/2)³ - 3(-1/2)² + 1 = -1/8 - 3/4 + 1 = 1/8
   • f(0) = 0³ - 3(0)² + 1 = 1
   • f(2) = 2³ - 3(2)² + 1 = 8 - 12 + 1 = -3
   • f(4) = 4³ - 3(4)² + 1 = 64 - 48 + 1 = 17
6. Identify Absolute Maximum and Minimum: The absolute maximum is 17 (at x = 4) and the absolute minimum is -3 (at x = 2).

Common Mistakes to Avoid

• Forgetting to Check Where f'(x) is Undefined: Many students only focus on finding where f'(x) = 0 and neglect to consider where the derivative might be undefined. This can lead to missing critical numbers.
• Not Checking the Domain of the Original Function: Critical numbers must be in the domain of the original function. Failing to check this can lead to incorrect conclusions.
• Confusing Critical Numbers with Critical Points: Remember that critical numbers are x-values. A critical point is a coordinate (c, f(c)).
• Assuming a Critical Number is Always a Local Extrema: A critical number indicates a potential local maximum or minimum. Further analysis (e.g., using the first or second derivative test) is needed to confirm its nature.
• Incorrectly Applying Differentiation Rules: A solid understanding of differentiation rules is essential. Mistakes in finding the derivative will lead to incorrect critical numbers.
• Algebraic Errors: Careful algebraic manipulation is crucial when solving for x after setting f'(x) = 0. Double-check your work to avoid errors.

Examples and Applications

Example 1: Finding Critical Numbers of a Polynomial Function

Let f(x) = x³ - 6x² + 5.

1. Find the Derivative: f'(x) = 3x² - 12x.
2. Find Where f'(x) = 0: 3x² - 12x = 0 => 3x(x - 4) = 0 => x = 0 or x = 4.
3. Find Where f'(x) is Undefined: The derivative is a polynomial, so it is defined everywhere.
4. Check the Domain: The domain is all real numbers.
5. Critical Numbers: x = 0 and x = 4.

Example 2: Finding Critical Numbers of a Rational Function

Let f(x) = (x²) / (x - 2).

1. Find the Derivative: Using the quotient rule: f'(x) = [(x - 2)(2x) - x²(1)] / (x - 2)² = (2x² - 4x - x²) / (x - 2)² = (x² - 4x) / (x - 2)².
2. Find Where f'(x) = 0: (x² - 4x) / (x - 2)² = 0 => x² - 4x = 0 => x(x - 4) = 0 => x = 0 or x = 4.
3. Find Where f'(x) is Undefined: The derivative is undefined when the denominator is zero: (x - 2)² = 0 => x = 2.
4. Check the Domain: The original function is undefined at x = 2. Therefore, x = 2 is not a critical number.
5. Critical Numbers: x = 0 and x = 4.

Example 3: Optimization Problem

A farmer wants to fence off a rectangular field bordering a straight river. He has 1000 feet of fencing. What dimensions of the field will maximize the enclosed area?

1. Define Variables: Let x be the length of the fence perpendicular to the river and y be the length of the fence parallel to the river.
2. Objective Function: We want to maximize the area A = xy.
3. Constraint: The total fencing used is 2x + y = 1000.
4. Express A in terms of one variable: Solve the constraint for y: y = 1000 - 2x. Substitute into the area equation: A(x) = x(1000 - 2x) = 1000x - 2x².
5. Find Critical Numbers: A'(x) = 1000 - 4x. Set A'(x) = 0: 1000 - 4x = 0 => x = 250.
6. Check Endpoints (practical domain): x must be between 0 and 500 (since y must be non-negative). If x=0 or x=500, the area is 0.
7. Determine Maximum Area: When x = 250, y = 1000 - 2(250) = 500. The maximum area is A = 250 * 500 = 125,000 square feet.

Therefore, the dimensions that maximize the enclosed area are x = 250 feet and y = 500 feet.

Advanced Techniques and Considerations

• Second Derivative Test: While the first derivative helps find critical numbers, the second derivative test can help determine whether a critical point is a local maximum or local minimum. If f''(c) > 0, then f(x) has a local minimum at x = c. If f''(c) < 0, then f(x) has a local maximum at x = c. If f''(c) = 0, the test is inconclusive.

• Implicit Differentiation: When dealing with implicitly defined functions, you'll need to use implicit differentiation to find dy/dx, and then proceed as usual to find critical numbers.

• Functions with Multiple Variables: The concept of critical points extends to functions of several variables. In this case, you need to find the partial derivatives with respect to each variable and set them equal to zero. Solving the resulting system of equations will give you the critical points. The nature of these critical points (local maxima, local minima, or saddle points) can be determined using the second partial derivative test.

Frequently Asked Questions (FAQ)

• What is the difference between a critical number and a stationary point?

  A stationary point is a point where the derivative is zero (f'(x) = 0). A critical number also includes points where the derivative is undefined. Therefore, all stationary points are critical points, but not all critical points are stationary points.

• Can a function have infinitely many critical numbers?

  Yes, some functions can have infinitely many critical numbers. For example, trigonometric functions like sin(x) and cos(x) have infinitely many critical numbers because their derivatives oscillate between -1 and 1, reaching 0 at infinitely many points.

• What does it mean if a function has no critical numbers?

  If a function has no critical numbers, it means that its derivative is never zero and never undefined within its domain. This implies that the function is either always increasing or always decreasing.

• Are endpoints always critical numbers?

  When considering a function on a closed interval, the endpoints are always considered when finding the absolute maximum and minimum values, even though they may not technically satisfy the definition of a critical number (i.e., the derivative is not necessarily zero or undefined at the endpoints).

• How do critical numbers relate to concavity?

  Critical numbers are related to the first derivative and tell us about increasing/decreasing intervals and potential local extrema. Concavity, on the other hand, is related to the second derivative and tells us about the rate of change of the slope. While not directly related, understanding both critical numbers and concavity provides a comprehensive view of a function's behavior.

Conclusion

Mastering the art of finding critical numbers is crucial for any calculus student or anyone working with mathematical models. This detailed guide has covered the definition, importance, step-by-step process, common mistakes, and advanced techniques associated with critical numbers. By diligently practicing these concepts and applying them to various problems, you can unlock a deeper understanding of functions and their behavior, and gain a powerful tool for solving optimization and analysis problems. Remember to always check for points where the derivative is undefined and verify that your critical numbers are within the domain of the original function. With consistent effort and a thorough understanding of these principles, you will be well-equipped to tackle even the most challenging calculus problems.