Finding Critical Numbers Of F(x) = X³ + 3x + 4
In calculus, identifying critical numbers is a fundamental step in understanding the behavior of a function. Critical numbers help us locate local maxima, local minima, and points where the function's rate of change is zero or undefined. This article delves into the process of finding the critical numbers of the function f(x) = x³ + 3x + 4. We'll explore the necessary calculus concepts, the step-by-step methodology, and the interpretation of the results. So, let's dive in and master the art of finding critical numbers!
Understanding Critical Numbers
Before we jump into the specifics of the given function, let's define what critical numbers are and why they are crucial in calculus.
Critical numbers, also known as critical points, are the values in the domain of a function f(x) where the derivative f'(x) is either equal to zero or is undefined. These points are significant because they potentially indicate where the function changes direction (from increasing to decreasing or vice versa), or where the function has a horizontal tangent line. In other words, critical numbers are the candidates for local maxima, local minima, or saddle points of the function.
To effectively locate critical numbers, it’s essential to first grasp the concept of the derivative of a function. The derivative f'(x) represents the instantaneous rate of change of the function f(x) with respect to its input variable x. Geometrically, the derivative at a particular point gives the slope of the tangent line to the function's graph at that point. A derivative of zero indicates a horizontal tangent, a key characteristic at local extrema.
Another crucial aspect to consider is where the derivative might be undefined. This often occurs at points where the function has a sharp turn, a vertical tangent, or a discontinuity. These points are also included in the set of critical numbers and must be carefully examined.
In summary, critical numbers are vital tools in calculus for:
- Identifying Local Extrema: Critical numbers help pinpoint potential maxima and minima of a function within a specific interval.
- Analyzing Function Behavior: By examining the sign of the derivative around critical numbers, we can determine where the function is increasing or decreasing.
- Sketching Function Graphs: Critical numbers provide key points that aid in accurately sketching the graph of a function.
Steps to Find Critical Numbers
The process of finding critical numbers can be broken down into a few straightforward steps:
- Find the Derivative: Calculate the derivative f'(x) of the given function f(x).
- Set the Derivative to Zero: Solve the equation f'(x) = 0 for x. The solutions are the critical numbers where the tangent line is horizontal.
- Identify Undefined Points: Determine any values of x for which f'(x) is undefined. These points are also critical numbers.
- Check Domain: Ensure that the critical numbers you've found are within the domain of the original function f(x).
By methodically following these steps, you can confidently identify all the critical numbers of a given function and use them to analyze its behavior.
Applying the Steps to f(x) = x³ + 3x + 4
Now, let's apply these steps to the function f(x) = x³ + 3x + 4 to find its critical numbers.
Step 1: Find the Derivative
The first step is to calculate the derivative of the function f(x) = x³ + 3x + 4. We will use the power rule of differentiation, which states that if f(x) = xⁿ, then f'(x) = nxⁿ⁻¹. Also, the derivative of a constant is zero. Applying these rules:
- The derivative of x³ is 3x².
- The derivative of 3x is 3.
- The derivative of 4 is 0.
Therefore, the derivative of f(x) is:
f'(x) = 3x² + 3
This derivative represents the slope of the tangent line to the graph of f(x) at any given point x. The next step involves finding the points where this slope is either zero or undefined.
Step 2: Set the Derivative to Zero
To find the critical numbers where the tangent line is horizontal, we need to solve the equation f'(x) = 0. This means setting our derivative 3x² + 3 equal to zero:
3x² + 3 = 0
To solve for x, we can follow these algebraic steps:
- Subtract 3 from both sides: 3x² = -3
- Divide both sides by 3: x² = -1
Now, we have the equation x² = -1. This equation asks us to find a real number that, when squared, results in -1. However, the square of any real number is always non-negative (either zero or positive). Therefore, there is no real solution to this equation.
This result is significant because it implies that the function f(x) = x³ + 3x + 4 does not have any critical numbers where its derivative is zero. In other words, there are no points on the graph of this function where the tangent line is horizontal.
Step 3: Identify Undefined Points
The next step in finding critical numbers is to determine if there are any values of x for which the derivative f'(x) is undefined. Our derivative is f'(x) = 3x² + 3. This is a polynomial function, and polynomial functions are defined for all real numbers.
To elaborate, a polynomial function consists of terms that are constants multiplied by non-negative integer powers of the variable. In our case, f'(x) = 3x² + 3 is a quadratic polynomial, which is a specific type of polynomial. Polynomials are known for their smooth and continuous behavior across the entire real number line.
Since f'(x) = 3x² + 3 is a polynomial, it does not have any points where it is undefined. There are no denominators that could be zero, no square roots of negative numbers, and no other operations that would lead to undefined values. This means that there are no critical numbers arising from points where the derivative is undefined.
Step 4: Check Domain
The final step in finding critical numbers is to ensure that the values we've identified (or, in this case, the lack thereof) are within the domain of the original function f(x) = x³ + 3x + 4. The domain of a function is the set of all possible input values (x-values) for which the function is defined.
Our original function, f(x) = x³ + 3x + 4, is also a polynomial function. As we discussed earlier, polynomial functions are defined for all real numbers. This means that any real number can be plugged into the function, and it will produce a valid output.
Since the domain of f(x) is all real numbers, we don't need to exclude any potential critical numbers based on domain restrictions. This is because we found that there are no critical numbers where f'(x) = 0 and no points where f'(x) is undefined. Therefore, the domain check does not introduce any additional constraints or considerations in this case.
Conclusion: Critical Numbers of f(x) = x³ + 3x + 4
After carefully following the steps to find critical numbers, we have reached the following conclusion:
The function f(x) = x³ + 3x + 4 has no critical numbers.
This means that there are no points on the graph of this function where the tangent line is horizontal, and there are no points where the derivative is undefined. This gives us valuable information about the behavior of the function. Since the derivative f'(x) = 3x² + 3 is always positive (because x² is always non-negative, and adding 3 makes the expression strictly positive), the function f(x) is always increasing. It has no local maxima, no local minima, and no points where it changes direction.
Implications and Interpretation
The absence of critical numbers for f(x) = x³ + 3x + 4 tells us a few important things about its graph and behavior:
- Monotonic Function: The function is strictly increasing over its entire domain. This means that as x increases, f(x) also increases, and there are no intervals where the function decreases.
- No Local Extrema: The function does not have any local maxima or local minima. This is because there are no points where the function changes from increasing to decreasing or vice versa.
- Smooth Curve: The graph of the function is a smooth, continuous curve without any sharp turns or vertical tangents.
By understanding these implications, we can sketch a rough graph of the function. It will be a curve that rises continuously from left to right, without any peaks or valleys. This is a classic example of how finding critical numbers (or the lack thereof) can significantly enhance our understanding of a function's behavior.
Choosing the Correct Answer
Based on our analysis, the correct answer to the question "What are the critical number(s) of f(x) = x³ + 3x + 4?" is:
(a) None
This is because we systematically found the derivative, set it equal to zero, checked for undefined points, and considered the domain. None of these steps yielded any critical numbers for the given function.
In summary, critical numbers are powerful tools in calculus for understanding the behavior of functions. By following a structured approach, we can confidently identify critical numbers and use them to analyze the increasing and decreasing intervals, local extrema, and overall shape of a function's graph. In the case of f(x) = x³ + 3x + 4, the absence of critical numbers revealed its strictly increasing nature and lack of local extrema.
