Finding Values Of K For Increasing Function F(x) = X³ - 3x² + Kx + 8

In the realm of calculus, understanding the behavior of functions is paramount. We often want to know where a function is increasing, decreasing, or remaining constant. This information is crucial for optimization problems, curve sketching, and a deeper understanding of the function's nature. In this article, we delve into the concept of increasing functions and apply it to a specific cubic polynomial. We aim to find the values of the constant k for which the function f(x) = x³ - 3x² + kx + 8 is an increasing function. This involves leveraging the power of differential calculus, specifically the first derivative test, to analyze the function's behavior.

What is an Increasing Function?

Before we dive into the specific problem, let's solidify our understanding of what constitutes an increasing function. A function f(x) is said to be increasing over an interval if, for any two points x₁ and x₂ in that interval, where x₁ < x₂, we have f(x₁) ≤ f(x₂). In simpler terms, as the input x increases, the output f(x) also increases (or at least stays the same). A strictly increasing function adheres to a slightly stronger condition: f(x₁) < f(x₂) for x₁ < x₂, meaning the output strictly increases as the input increases. Geometrically, an increasing function's graph rises as we move from left to right.

The Role of the Derivative

The derivative of a function, denoted as f'(x), plays a pivotal role in determining the intervals where the function is increasing or decreasing. The derivative represents the instantaneous rate of change of the function at a given point. A positive derivative indicates that the function is increasing at that point, a negative derivative indicates decreasing behavior, and a zero derivative suggests a stationary point (a local maximum, local minimum, or a saddle point).

The First Derivative Test provides a formal framework for using the derivative to identify intervals of increasing and decreasing behavior. It states that:

• If f'(x) > 0 for all x in an interval, then f(x) is increasing on that interval.
• If f'(x) < 0 for all x in an interval, then f(x) is decreasing on that interval.
• If f'(x) = 0 for all x in an interval, then f(x) is constant on that interval.

This test forms the cornerstone of our approach to solving the problem at hand.

Applying the Concepts to f(x) = x³ - 3x² + kx + 8

Now, let's apply our understanding to the given function, f(x) = x³ - 3x² + kx + 8. Our goal is to find the values of k for which this function is increasing. To achieve this, we will follow these steps:

1. Find the derivative f'(x): We begin by finding the derivative of the function using the power rule of differentiation.
2. Set f'(x) > 0: For the function to be increasing, its derivative must be positive. We set up the inequality f'(x) > 0.
3. Solve the inequality: We solve the resulting inequality to find the values of x for which f'(x) > 0. The solution will likely depend on the parameter k.
4. Determine the values of k: Finally, we analyze the solution to the inequality to determine the values of k that ensure f'(x) > 0 for all x (or at least over a significant interval), thus making the function increasing.

Step-by-Step Solution

Let's execute the steps outlined above:

1. Find the derivative f'(x)

Using the power rule of differentiation, we find the derivative of f(x) = x³ - 3x² + kx + 8:

f'(x) = 3x² - 6x + k

This derivative represents the slope of the tangent line to the curve of f(x) at any point x.

2. Set f'(x) > 0

For f(x) to be an increasing function, we need f'(x) > 0. Therefore, we set up the following inequality:

3x² - 6x + k > 0

3. Solve the inequality

To solve this quadratic inequality, we can analyze the discriminant of the quadratic expression. The discriminant, denoted as Δ, is given by:

Δ = b² - 4ac

where a, b, and c are the coefficients of the quadratic expression ax² + bx + c. In our case, a = 3, b = -6, and c = k. Therefore, the discriminant is:

Δ = (-6)² - 4(3)(k) = 36 - 12k

The sign of the discriminant tells us about the nature of the roots of the quadratic equation 3x² - 6x + k = 0:

• If Δ > 0, the equation has two distinct real roots.
• If Δ = 0, the equation has one real root (a repeated root).
• If Δ < 0, the equation has no real roots.

For the inequality 3x² - 6x + k > 0 to hold for all x, the parabola represented by the quadratic expression must open upwards (which it does since a = 3 > 0) and have no real roots. This means the discriminant must be negative:

Δ < 0

36 - 12k < 0

4. Determine the values of k

Now, we solve the inequality for k:

36 < 12k

k > 3

Therefore, the values of k for which f(x) = x³ - 3x² + kx + 8 is an increasing function are k > 3.

Verification and Interpretation

We can verify this result by considering a few cases:

• Case 1: k = 4 (k > 3) If k = 4, then f'(x) = 3x² - 6x + 4. The discriminant is 36 - 12(4) = -12, which is negative. This means f'(x) is always positive, and f(x) is increasing.
• Case 2: k = 3 If k = 3, then f'(x) = 3x² - 6x + 3 = 3(x - 1)². f'(x) is greater than or equal to 0, so the function f(x) is non-decreasing. While it's not strictly increasing everywhere (it has a stationary point at x=1), it does not decrease.
• Case 3: k = 2 (k < 3) If k = 2, then f'(x) = 3x² - 6x + 2. The discriminant is 36 - 12(2) = 12, which is positive. This means f'(x) has two real roots, and f(x) will have intervals where it is increasing and decreasing.

This confirms our solution that k > 3 is the condition for f(x) to be an increasing function.

Conclusion

In this article, we successfully determined the values of k for which the cubic function f(x) = x³ - 3x² + kx + 8 is an increasing function. We achieved this by applying the concept of the first derivative test and analyzing the discriminant of the quadratic derivative. The key finding is that the function is increasing when k > 3. This problem illustrates the powerful connection between differential calculus and the analysis of function behavior. By understanding the derivative and its relationship to increasing and decreasing intervals, we can gain valuable insights into the properties of functions and their applications in various fields.

1. How do you determine if a function is increasing?

To determine if a function is increasing, you can use the first derivative test. Find the derivative of the function, f'(x). If f'(x) > 0 over an interval, the function is increasing on that interval. If f'(x) ≥ 0, the function is non-decreasing.

2. What is the difference between an increasing function and a strictly increasing function?

An increasing function f(x) satisfies the condition f(x₁) ≤ f(x₂) for x₁ < x₂. This means the function can either increase or stay constant. A strictly increasing function, on the other hand, satisfies the stricter condition f(x₁) < f(x₂) for x₁ < x₂. This means the function's output must strictly increase as the input increases.

3. Can a function be increasing and decreasing in different intervals?

Yes, a function can be increasing in some intervals and decreasing in others. The derivative of the function determines these intervals. Where f'(x) > 0, the function is increasing, and where f'(x) < 0, the function is decreasing.

4. How does the discriminant relate to increasing functions?

The discriminant of a quadratic expression (like a derivative) helps determine the nature of the roots. In the context of increasing functions, if the derivative is a quadratic expression, a negative discriminant implies that the derivative has no real roots and maintains the same sign (either always positive or always negative). For a function to be increasing, we often want the derivative to be always positive, which can be ensured by a negative discriminant (if the quadratic opens upwards).

5. What are some real-world applications of increasing functions?

Increasing functions have numerous real-world applications. They can model:

• Population growth: As time increases, the population typically increases (assuming birth rate exceeds death rate).
• Compound interest: As time passes, the amount of money in an account with compound interest increases.
• The relationship between input and output in a production process: In many cases, increasing the input (e.g., labor or raw materials) leads to an increase in output.
• The price of an asset over time (although this is not always strictly increasing).

To further enhance your understanding of increasing functions and related concepts, consider exploring the following resources:

• Calculus textbooks: Standard calculus textbooks provide comprehensive explanations and examples of increasing functions, derivatives, and the first derivative test.
• Online calculus courses: Platforms like Coursera, edX, and Khan Academy offer excellent calculus courses that cover these topics in detail.
• Interactive calculus applets: Websites like Desmos and GeoGebra have interactive applets that allow you to visualize functions, derivatives, and their relationships.
• Practice problems: Work through a variety of practice problems to solidify your understanding and build problem-solving skills.

By delving deeper into these resources, you can develop a strong foundation in calculus and its applications, including the analysis of increasing and decreasing functions.