How To Find The Critical Number

Finding critical numbers is a cornerstone of calculus, providing the key to unlocking maximum and minimum values of a function, which are crucial in optimization problems across various fields. These values represent points where the function's behavior changes, indicating potential turning points, peaks, or valleys. Understanding how to find critical numbers equips you with a powerful tool for analyzing functions and solving real-world problems.

What Are Critical Numbers?

At its core, a critical number of a function, f(x), is a value, c, in the domain of f where either the derivative of f at c is equal to zero (f'(c) = 0) or the derivative of f at c is undefined (f'(c) does not exist). These points are significant because they often correspond to local maxima, local minima, or saddle points on the graph of the function.

Steps to Find Critical Numbers

The process of finding critical numbers involves a few key steps, each building upon the previous one:

Find the Derivative of the Function: The first step is to determine the derivative of the given function, f(x). This usually involves applying the rules of differentiation, such as the power rule, product rule, quotient rule, and chain rule, as appropriate.
Set the Derivative Equal to Zero: Once you have the derivative, f'(x), set it equal to zero and solve for x. The solutions to this equation are the values of x where the tangent line to the graph of f(x) is horizontal.
Find Where the Derivative is Undefined: In addition to finding where the derivative is zero, you also need to identify any values of x for which the derivative is undefined. This typically occurs when the derivative involves a fraction and the denominator is zero, or when dealing with functions that have sharp corners or vertical tangents.
Check If Critical Numbers are in the Domain of the Function: After you've identified the potential critical numbers, you must ensure they are actually in the domain of the original function, f(x). A value that makes the derivative zero or undefined but is not in the domain of the original function is not a critical number.
List the Critical Numbers: Compile all the values of x that satisfy the conditions above. These are the critical numbers of the function f(x).

Example 1: Finding Critical Numbers of a Polynomial Function

Let's find the critical numbers of the function f(x) = x³ - 6x² + 5.

Find the Derivative: Using the power rule, the derivative of f(x) is: f'(x) = 3x² - 12x.
Set the Derivative Equal to Zero: Set f'(x) = 0: 3x² - 12x = 0.
Solve for x: Factor out a 3x: 3x(x - 4) = 0. This gives us two solutions: x = 0 and x = 4.
Find Where the Derivative is Undefined: The derivative f'(x) = 3x² - 12x is a polynomial and is defined for all real numbers. Therefore, there are no values of x for which the derivative is undefined.
Check If Critical Numbers are in the Domain of the Function: Both x = 0 and x = 4 are in the domain of f(x) = x³ - 6x² + 5, which is all real numbers.
List the Critical Numbers: The critical numbers of f(x) = x³ - 6x² + 5 are x = 0 and x = 4.

Example 2: Finding Critical Numbers of a Rational Function

Let's find the critical numbers of the function f(x) = (x²)/(x - 2).

Find the Derivative: Using the quotient rule, the derivative of f(x) is:

f'(x) = [(x - 2)(2x) - x²(1)] / (x - 2)² f'(x) = (2x² - 4x - x²) / (x - 2)² f'(x) = (x² - 4x) / (x - 2)²
Set the Derivative Equal to Zero: Set f'(x) = 0: (x² - 4x) / (x - 2)² = 0. This is equivalent to x² - 4x = 0.
Solve for x: Factor out an x: x(x - 4) = 0. This gives us two solutions: x = 0 and x = 4.
Find Where the Derivative is Undefined: The derivative f'(x) = (x² - 4x) / (x - 2)² is undefined when the denominator is zero, which occurs at x = 2.
Check If Critical Numbers are in the Domain of the Function: The original function f(x) = (x²)/(x - 2) is undefined at x = 2, so x = 2 is not in the domain. The values x = 0 and x = 4 are in the domain of f(x).
List the Critical Numbers: The critical numbers of f(x) = (x²)/(x - 2) are x = 0 and x = 4.

Example 3: Finding Critical Numbers of a Function with a Radical

Let's find the critical numbers of the function f(x) = √(4 - x²).

Find the Derivative: Rewrite the function as f(x) = (4 - x²)^(1/2). Using the chain rule, the derivative of f(x) is:

f'(x) = (1/2)(4 - x²)^(-1/2) * (-2x) f'(x) = -x / √(4 - x²)
Set the Derivative Equal to Zero: Set f'(x) = 0: -x / √(4 - x²) = 0. This is equivalent to -x = 0, which gives us x = 0.
Find Where the Derivative is Undefined: The derivative f'(x) = -x / √(4 - x²) is undefined when the denominator is zero, which occurs when 4 - x² = 0. This gives us x = ±2.
Check If Critical Numbers are in the Domain of the Function: The original function f(x) = √(4 - x²) is defined only when 4 - x² ≥ 0, which means -2 ≤ x ≤ 2. Thus, the domain of f(x) is [-2, 2]. The value x = 0 is in the domain, but x = ±2 are also in the domain and make the derivative undefined, so they must be included as critical numbers.
List the Critical Numbers: The critical numbers of f(x) = √(4 - x²) are x = -2, 0, 2.

Why Are Critical Numbers Important?

Critical numbers play a vital role in determining the local extrema (maxima and minima) of a function. These extrema are crucial in various applications, such as:

Optimization Problems: Many real-world problems involve finding the maximum or minimum value of a function subject to certain constraints. Critical numbers help identify potential solutions. For example, maximizing profit, minimizing cost, or optimizing resource allocation.
Curve Sketching: Critical numbers help in sketching the graph of a function by indicating where the function is increasing or decreasing and where it has local extrema. They are also used to determine the concavity of the curve.
Related Rates Problems: In related rates problems, where we need to find the rate of change of one quantity in terms of the rate of change of another, critical numbers can help identify when a rate is maximized or minimized.
Physics and Engineering: Critical points are essential in analyzing systems in physics and engineering, such as finding equilibrium points or determining the stability of a system.

Techniques for Finding Where the Derivative is Undefined

Identifying where the derivative of a function is undefined is just as crucial as finding where it equals zero. Here are several common scenarios and techniques for handling them:

Rational Functions: If the derivative is a rational function (a fraction), it is undefined when the denominator equals zero. Set the denominator equal to zero and solve for x. However, always remember to check if these x values are also in the domain of the original function.
Radical Functions: If the derivative involves a square root or any even root, it is undefined when the expression inside the root is negative. Additionally, if the radical is in the denominator, the derivative is undefined when the expression inside the root is zero. Make sure that the x values you find are in the domain of the original function.
Absolute Value Functions: The derivative of an absolute value function is often undefined at points where the expression inside the absolute value is zero. This is because absolute value functions have sharp corners at these points.
Piecewise Functions: For piecewise functions, the derivative may be undefined at the points where the function definition changes. You need to evaluate the derivative from the left and right sides of these points to determine if the derivative exists.
Trigonometric Functions: For trigonometric functions, be mindful of points where functions like tangent or cotangent are undefined (multiples of π/2 for tangent and multiples of π for cotangent).

Using the First Derivative Test

Once you have found the critical numbers of a function, you can use the first derivative test to determine whether each critical number corresponds to a local maximum, a local minimum, or neither.

The first derivative test involves evaluating the sign of the derivative f'(x) to the left and right of each critical number c. There are three possible outcomes:

Local Minimum: If f'(x) changes from negative to positive at x = c, then f(x) has a local minimum at x = c. This means the function is decreasing to the left of c and increasing to the right of c.
Local Maximum: If f'(x) changes from positive to negative at x = c, then f(x) has a local maximum at x = c. This means the function is increasing to the left of c and decreasing to the right of c.
Neither: If f'(x) does not change sign at x = c, then f(x) has neither a local minimum nor a local maximum at x = c. This indicates a saddle point or an inflection point where the concavity changes.

Example: Applying the First Derivative Test

Let's apply the first derivative test to the function f(x) = x³ - 6x² + 5, which we found to have critical numbers at x = 0 and x = 4. The derivative is f'(x) = 3x² - 12x = 3x(x - 4).

Analyze the Sign of f'(x) around x = 0:
- For x < 0, let's choose x = -1: f'(-1) = 3(-1)(-1 - 4) = 15 > 0 (positive).
- For x > 0, let's choose x = 1: f'(1) = 3(1)(1 - 4) = -9 < 0 (negative).
Since f'(x) changes from positive to negative at x = 0, f(x) has a local maximum at x = 0.
Analyze the Sign of f'(x) around x = 4:
- For x < 4, let's choose x = 3: f'(3) = 3(3)(3 - 4) = -9 < 0 (negative).
- For x > 4, let's choose x = 5: f'(5) = 3(5)(5 - 4) = 15 > 0 (positive).
Since f'(x) changes from negative to positive at x = 4, f(x) has a local minimum at x = 4.

Therefore, f(x) = x³ - 6x² + 5 has a local maximum at x = 0 and a local minimum at x = 4.

Common Mistakes to Avoid

When finding critical numbers, it's important to avoid some common mistakes:

Forgetting to Check the Domain: Always check if the potential critical numbers are in the domain of the original function. A value that is not in the domain cannot be a critical number.
Only Finding Where the Derivative is Zero: Make sure to also find where the derivative is undefined. These points can also be critical numbers.
Algebra Mistakes: Be careful with algebraic manipulations when finding and solving for the derivative. A small mistake can lead to incorrect critical numbers.
Misapplying Differentiation Rules: Ensure you correctly apply the differentiation rules (power rule, product rule, quotient rule, chain rule) when finding the derivative.

Applications in Real-World Problems

Finding critical numbers has numerous applications in real-world problems across various fields. Here are a few examples:

Economics: A company wants to maximize its profit. The profit function is given by P(x) = R(x) - C(x), where R(x) is the revenue function and C(x) is the cost function. To find the production level x that maximizes profit, we find the critical numbers of P(x).
Engineering: An engineer wants to design a cylindrical container that minimizes the surface area while maintaining a fixed volume. The surface area A is a function of the radius r and height h, and the volume V is fixed. By expressing A as a function of one variable and finding its critical numbers, the engineer can determine the optimal dimensions.
Physics: A projectile is launched into the air, and its height h(t) at time t is given by a quadratic function. To find the maximum height reached by the projectile, we find the critical number of h(t).
Computer Science: In machine learning, optimization algorithms are used to minimize a cost function. Finding critical points (where the gradient is zero) is crucial for these algorithms to converge to an optimal solution.

Advanced Techniques and Considerations

While the basic steps for finding critical numbers are straightforward, some functions require more advanced techniques and considerations:

Implicit Differentiation: For functions defined implicitly, such as x² + y² = 25, you need to use implicit differentiation to find the derivative. Then, set the derivative equal to zero or find where it is undefined to find the critical numbers.
Multivariable Functions: For functions of multiple variables, such as f(x, y), you need to find partial derivatives with respect to each variable. The critical points are the solutions to the system of equations where all partial derivatives are equal to zero.
Constrained Optimization: In some problems, you need to find the critical numbers of a function subject to a constraint. Techniques like Lagrange multipliers can be used to solve these problems.

Conclusion

Mastering the technique of finding critical numbers is fundamental to understanding and applying calculus in diverse fields. By systematically following the steps—finding the derivative, setting it equal to zero or undefined, and checking the domain—you can unlock the potential of functions to reveal their maximum and minimum values. These values are not just theoretical curiosities but powerful tools for solving optimization problems and gaining insights into real-world phenomena. Embrace the practice, avoid common mistakes, and watch as your ability to analyze and interpret functions deepens, leading to a richer understanding of the world around you.