Question

What are the conditions for a critical point of a function \(f ?\)

Short answer

Answer: The key steps to find the critical points of a function f(x) are: 1) Find the derivative of the function (f'(x)), 2) Set the derivative equal to zero and solve for x, 3) Determine where the derivative is undefined, and 4) Combine the results to get the complete set of critical points.

Step-by-step solution

Find the derivative of the function

To find the critical points of a function, we first need to find its derivative, which represents the slope of the tangent line to the function's curve. The derivative of a function f(x) is denoted as f'(x) or df/dx.

Set the derivative equal to zero

Once we have found the derivative of the function, we need to set it equal to zero. This will allow us to find the points (x-values) where the function has a horizontal tangent line, indicating a possible local maximum or minimum.

Solve for the critical points

Now, solve the equation f'(x)=0 for x to find the critical points. These points are the x-coordinates where the tangent line to the curve has a slope of zero, indicating a potential local maximum or minimum.

Determine where the derivative is undefined

Additionally, we need to determine where the derivative f'(x) is undefined. These are the points where the function has a vertical tangent line or a discontinuity in the tangent line's slope.

Combine the results

Combine the x-coordinates where the derivative is zero and where the derivative is undefined to find the complete set of critical points. In summary, the critical points of a function are the points where the derivative is either zero or undefined. By following these steps, one can find the conditions for a critical point of a given function f(x).

Key concepts

Finding Derivatives

Understandably, the process of finding derivatives is quintessential in calculus when we're discussing the behavior of functions. The derivative of a function represents the rate of change of the function's value with respect to changes in its input value, typically denoted as variable 'x'. To find the derivative, you apply rules of differentiation such as the power rule, product rule, quotient rule, and the chain rule, depending on the form of the function.
In practical terms, when you calculate the derivative of function, say, f(x), you’re determining the formula for the slope of the function's curve at any point 'x'. This action sets the stage for identifying the function's critical points which are linked to its local minima and maxima—areas of high importance in many fields, such as economics and engineering for optimizing resources.

Setting Derivatives to Zero

To find where a function's graph has horizontal tangent lines—which is one of the markers for potential local maxima or minima—we set the derivative of the function equal to zero. This operation is rooted in the graphical interpretation of a derivative as the slope of the tangent line. A slope of zero corresponds to a horizontal line, meaning the function’s rate of change at that point is zero.
By solving the equation f'(x) = 0, you determine the 'x' values at which the function's curve is flat (neither increasing nor decreasing). This is only an indication, not a confirmation, that a maximum, minimum, or saddle point may exist at these 'x' values. These points where the first derivative equals zero are called stationary points.

Solving for x

Having set the derivative of the function to zero, the next task is 'solving for x'. This means isolating 'x' in the equation f'(x) = 0 to determine the actual numeric values of the potential critical points. Depending on the complexity of the derivative, you may apply different algebraic techniques such as factoring, using the quadratic formula, or employing more advanced methods like Newton's method if the equation cannot be easily solved by hand.
It's important to consider all possible solutions, including negative numbers, zero, and positive numbers, since the function's behavior can significantly vary across its domain. Doing so provides a list of candidate x-values where the function might have a local maximum, minimum, or other critical features.

Undefined Derivatives

Apart from where the derivative equals zero, critical points of a function can also occur where the derivative is undefined. These are points where the function could have a corner, cusp, vertical tangent, or discontinuity. For a function f(x), such points are usually found where a denominator is zero (in the case of rational functions) or where a function involves a root of a variable in which taking a derivative isn't straightforward.
Identifying these points requires careful analysis of the derivative formula. They can sometimes signify a break in the smoothness of the function, and understanding their existence is crucial to fully describing the function's behavior across its domain.

Local Maximum and Minimum

Once the potential critical points have been found by setting the derivative to zero and identifying points where the derivative is undefined, it's necessary to figure out whether these points are local maxima, minima, or neither. This determination can be made using various tests, such as the First Derivative Test or the Second Derivative Test.
A local maximum occurs at a point x=c if f(c) is greater than any other value of f(x) close to c. Conversely, a local minimum occurs at x=c if f(c) is less than any other value of f(x) nearby. These points are crucial for understanding the overall behavior of the function as they represent the peaks and valleys of the function's graph, and they can often provide practical insights for optimization problems.