Problem 21 - Calculus A New Horizon

Question

Use any method to find the relative extrema of the function \(f\). $$f(x)=x^{3}+5 x-2$$

Step-by-Step Solution

Answer

No relative extrema exist for the function.

Step 1: Find the Derivative of the Function

To find the relative extrema of a function, first compute the derivative of the function. The derivative, denoted as \(f'(x)\), of the function \(f(x) = x^3 + 5x - 2\) is calculated using the power rule. Therefore, \(f'(x) = 3x^2 + 5\).

Step 2: Set the Derivative Equal to Zero

Set the derivative \(f'(x)\) equal to zero to find the critical points where the function could have relative extrema. This results in the equation: \[3x^2 + 5 = 0\].

Step 3: Solve for Critical Points

Solve the equation \(3x^2 + 5 = 0\) for \(x\). Since the sum of a positive term \(3x^2\) and a positive constant \(5\) can never be zero, the equation has no real solutions. Thus, there are no critical points.

Step 4: Conclude About Relative Extrema

Since there are no critical points derived from \(3x^2 + 5 = 0\), the function \(f(x) = x^3 + 5x - 2\) has no relative extrema, because relative extrema occur at critical points where the derivative is either zero or undefined. Here, no such points exist.

Key Concepts

Derivative

Derivatives play a crucial role in calculus as they help us understand how functions change. At its core, a derivative tells us the rate at which a function is increasing or decreasing. More formally, the derivative of a function at a particular point gives us the slope of the tangent line to the graph of the function at that point. For example, if you have a function like \( f(x) = x^3 + 5x - 2 \), you use the rules of differentiation to find its derivative. In this case, using the power rule, the derivative is \( f'(x) = 3x^2 + 5 \). Key points about derivatives:

• The derivative provides critical information about the function’s behavior.
• Using rules like the power rule simplifies the process of finding derivatives.

Once the derivative is found, it helps in locating critical points, which are essential in analyzing the function’s relative extrema.

Relative Extrema

Relative extrema refer to the points in the domain of a function where the function takes a relative maximum or minimum value. These points are often described as "peaks" or "valleys" of the graph. Determining these values is critical for understanding the overall behavior of the function. To find these extrema:

• First, compute the derivative of the function.
• Set this derivative equal to zero to find the critical points.

Consider the function \( f(x) = x^3 + 5x - 2 \). After finding the derivative \( f'(x) = 3x^2 + 5 \), you equate it to zero: \[3x^2 + 5 = 0\] This equation would typically help locate relative extrema. However, if it yields no real solutions, as is the case here, it indicates that the function has no relative maximum or minimum values. Therefore, no "peaks" or "valleys" exist on the graph.

Critical Points

Critical points are invaluable tools in calculus for identifying where a function's graph might change direction. These points occur where the derivative of the function is zero or undefined. They serve as potential locations for relative extrema, which greatly influence the graph's shape. For the function \( f(x) = x^3 + 5x - 2 \), we derived \( f'(x) = 3x^2 + 5 \). By setting this derivative equal to zero, \[ 3x^2 + 5 = 0 \] one would normally solve for \( x \) to find where critical points occur. However, this equation offers no real numbers, indicating a lack of critical points. In such cases, the function doesn't switch from increasing to decreasing at any real number in its domain, highlighting the absence of relative extrema. In conclusion, without critical points, understanding the behavior of the function in terms of turning points remains limited.