What Is The Maximum Possible Number Of Extreme Values For The Function F(x) = X³ + 4x² - 3x - 10?
Understanding Extreme Values
In calculus, extreme values, also known as extrema, refer to the maximum and minimum values of a function. These points are crucial in understanding the behavior of a function and are essential in various applications, including optimization problems. The extreme values of a function can be classified into two main types: local extrema (also known as relative extrema) and global extrema (also known as absolute extrema). Local extrema represent the maximum or minimum value of a function within a specific interval, while global extrema represent the overall maximum or minimum value of the function over its entire domain. To find these extreme values, we often use derivatives, which provide information about the rate of change of the function.
The relationship between the derivative of a function and its extreme values is fundamental in calculus. Specifically, critical points, where the derivative is either zero or undefined, are potential locations for local extrema. By analyzing the sign of the first derivative around these critical points, we can determine whether the function is increasing or decreasing, and thus identify local maxima and local minima. Additionally, the second derivative test can be used to further confirm the nature of these critical points. A positive second derivative indicates a local minimum, while a negative second derivative indicates a local maximum. Understanding these concepts is crucial for solving optimization problems and analyzing the behavior of functions in various contexts.
Polynomial functions, in particular, exhibit specific characteristics regarding their extreme values. A polynomial function of degree n can have at most n-1 extreme values. This is because the derivative of a polynomial function of degree n is a polynomial function of degree n-1, and the roots of this derivative correspond to the critical points of the original function. Therefore, the number of real roots of the derivative, which are potential locations for extreme values, is limited by the degree of the derivative polynomial. This property is highly useful in analyzing the behavior of polynomial functions and determining the maximum possible number of local maxima and local minima. When dealing with higher-degree polynomials, it's essential to remember this relationship to efficiently find and classify extreme values.
Analyzing the Given Function: f(x) = x³ + 4x² - 3x - 10
To determine the maximum number of possible extreme values for the given function, f(x) = x³ + 4x² - 3x - 10, we need to analyze its derivative. The function is a polynomial of degree 3, which is a cubic function. Understanding the properties of polynomial functions is crucial here. As discussed earlier, a polynomial function of degree n can have at most n-1 extreme values. This is a fundamental concept in calculus and helps us predict the maximum possible number of local maxima and local minima a function can have.
To find the extreme values, we first need to find the first derivative of the function. The first derivative, denoted as f'(x), represents the rate of change of the function and helps us identify critical points. The derivative of f(x) = x³ + 4x² - 3x - 10 is calculated using the power rule, which states that the derivative of xⁿ is nxⁿ⁻¹. Applying this rule to each term in the function, we get f'(x) = 3x² + 8x - 3. This quadratic equation is crucial for finding the critical points of the original function.
The critical points of a function are the points where the first derivative is either equal to zero or undefined. In this case, f'(x) = 3x² + 8x - 3 is a quadratic equation, and we need to find the values of x for which f'(x) = 0. Setting the derivative equal to zero, we have 3x² + 8x - 3 = 0. Solving this quadratic equation will give us the x-coordinates of the critical points. These critical points are potential locations for local maxima and local minima. The number of real solutions to this equation will determine the number of critical points, and consequently, the maximum number of extreme values.
Finding the Derivative and Critical Points
As established, finding the derivative of the function f(x) = x³ + 4x² - 3x - 10 is the first step in determining its extreme values. The derivative, f'(x), helps us identify the critical points, which are the potential locations of local maxima and local minima. We calculated the derivative using the power rule, which is a fundamental concept in calculus. The result of this calculation is f'(x) = 3x² + 8x - 3. This quadratic equation is crucial for our analysis.
To find the critical points, we need to solve the equation f'(x) = 0. This means we need to find the values of x that satisfy the equation 3x² + 8x - 3 = 0. There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and completing the square. In this case, we can factor the quadratic equation as (3x - 1)(x + 3) = 0. Factoring is often the quickest method if the quadratic equation can be easily factored.
By setting each factor equal to zero, we find the solutions for x. From (3x - 1) = 0, we get x = 1/3. From (x + 3) = 0, we get x = -3. These two values, x = 1/3 and x = -3, are the critical points of the function f(x). These are the x-coordinates where the function's slope is zero, indicating potential extreme values. The fact that we found two distinct real roots for the derivative equation confirms that the original cubic function can have at most two extreme values.
Determining the Maximum Number of Extreme Values
Having found the critical points, x = 1/3 and x = -3, we now determine the maximum number of extreme values the function f(x) = x³ + 4x² - 3x - 10 can have. The critical points are potential locations for local maxima and local minima. However, it is important to remember that not every critical point corresponds to an extreme value. To confirm whether a critical point is a local maximum, a local minimum, or neither, we can use either the first derivative test or the second derivative test.
The first derivative test involves analyzing the sign of the first derivative, f'(x), around the critical points. If f'(x) changes sign from positive to negative at a critical point, it indicates a local maximum. Conversely, if f'(x) changes sign from negative to positive, it indicates a local minimum. If f'(x) does not change sign, the critical point is neither a local maximum nor a local minimum. The second derivative test involves evaluating the second derivative, f''(x), at the critical points. If f''(x) > 0, the critical point is a local minimum. If f''(x) < 0, the critical point is a local maximum. If f''(x) = 0, the test is inconclusive.
In this case, since we have a cubic function, we know it can have at most two extreme values. A cubic function can have a local maximum and a local minimum, or it can have neither. Our calculations have shown that the derivative, f'(x) = 3x² + 8x - 3, has two distinct real roots, meaning there are two critical points. Therefore, the maximum number of possible extreme values for the function f(x) = x³ + 4x² - 3x - 10 is 2. This aligns with the general rule that a polynomial of degree n can have at most n-1 extreme values.
Conclusion: The Answer is C. 2
In conclusion, after analyzing the function f(x) = x³ + 4x² - 3x - 10, we determined that the maximum number of possible extreme values is 2. This was achieved by first finding the derivative of the function, f'(x) = 3x² + 8x - 3, and then solving for the critical points by setting the derivative equal to zero. The solutions, x = 1/3 and x = -3, represent the critical points, which are potential locations for local maxima and local minima.
Given that the function is a cubic polynomial (degree 3), it can have at most 3-1 = 2 extreme values. The two critical points we found confirm that this is indeed the maximum possible number of extreme values for this function. While we did not explicitly determine whether these critical points are local maxima or local minima, the question only asked for the maximum number of possible extreme values, which we have successfully found.
Therefore, the correct answer is C. 2. This problem highlights the importance of understanding the relationship between the derivative of a function and its extreme values, as well as the properties of polynomial functions. By applying these concepts, we can efficiently determine the maximum number of local maxima and local minima a function can have.