How do you find the critical value of a function?

As an expert in the field of mathematics, particularly in calculus, I can guide you through the process of finding the critical values of a function. Critical values are points on the graph of a function where the derivative is either zero or undefined. These points are significant because they often correspond to local maxima, local minima, or points of inflection. Here's a step-by-step guide on how to find the critical values: ### Step 1: Find the Derivative of the Function The first step in finding critical points is to find the derivative of the function. The derivative, denoted as \( f'(x) \) or \( \frac{d}{dx}f(x) \), represents the rate of change of the function at any given point. It can be found using various differentiation rules such as the power rule, product rule, quotient rule, or chain rule, depending on the complexity of the function. ### Step 2: Set the Derivative Equal to Zero and Solve for \( x \) Once you have the derivative, the next step is to set it equal to zero and solve for \( x \). This is because critical points occur where the derivative is zero (or undefined), indicating a change in the function's behavior. The solutions to \( f'(x) = 0 \) are known as critical numbers. ### Step 3: Determine if the Points are Maxima, Minima, or Points of Inflection After finding the critical numbers, you need to determine whether each corresponds to a local maximum, local minimum, or a point of inflection. This can be done using the following methods: - First Derivative Test: Examine the sign of the derivative to the left and right of each critical number. If the derivative changes from positive to negative, you have a local maximum. If it changes from negative to positive, you have a local minimum. If there's no change in sign, the test is inconclusive. - Second Derivative Test: Find the second derivative, \( f''(x) \), and evaluate it at the critical points. If \( f''(x) > 0 \), the point is a local minimum. If \( f''(x) < 0 \), the point is a local maximum. If \( f''(x) = 0 \) or is undefined, the test is inconclusive. - Graphical Analysis: Sometimes, a graph of the function can provide insight into the nature of the critical points. ### Step 4: Find the Corresponding \( y \) Values Once you've identified the critical points, plug each \( x \) value back into the original function to find the corresponding \( y \) values. These \( (x, y) \) pairs are the critical points of the function. ### Step 5: Verify the Results It's always a good idea to verify your results by checking your work and ensuring that you haven't missed any critical points or made any errors in your calculations. ### Example Let's consider a simple example to illustrate the process: Given the function \( f(x) = x^2 - 4x \), we want to find the critical points. 1. Find the Derivative: \( f'(x) = 2x - 4 \) 2. Set the Derivative Equal to Zero: \( 2x - 4 = 0 \) leads to \( x = 2 \) (critical number). 3. Determine the Nature of the Point: Since \( f'(x) \) changes from negative to positive around \( x = 2 \), we have a local minimum. 4. Find the Corresponding \( y \) Value: \( f(2) = 2^2 - 4(2) = 4 - 8 = -4 \), so the critical point is \( (2, -4) \). ### Conclusion Finding critical values is a fundamental part of calculus that helps us understand the behavior of functions. By following these steps, you can systematically find and analyze critical points to gain insights into the function's local and global behavior.

To find these critical points you must first take the derivative of the function. Second, set that derivative equal to 0 and solve for x. Each x value you find is known as a critical number. Third, plug each critical number into the original equation to obtain your y values.