Problem 69
Question
Find the derivative of the given function \(f\). Then use a graphing utility to graph \(f\) and its derivative in the same viewing window. What does the \(x\) -intercept of the derivative indicate about the graph of \(f ?\) $$ f(x)=x^{3}-3 x $$
Step-by-Step Solution
Verified Answer
The derivative of the function \(f(x)=x^{3}-3x\) is \(f'(x)=3x^{2}-3\). The derivative has x-intercepts at \(x = ±1\) indicating the stationary points of the original function. On the graph, these points represent local minima and maxima.
1Step 1: Find the Derivative
Firstly, apply the power rule to the function \(f(x)=x^{3}-3x\). The power rule states that the derivative of \(x^{n}\) is \(nx^{n-1}\). As such, the derivative of \(f(x)=x^{3}-3x\) is \(f'(x)=3x^{2}-3\)
2Step 2: Graph the function and its derivative
Using a graphing utility, graph the function and its derivative. The function \(f(x)=x^{3}-3x\) is a cubic function and its graph will be a curve. The derivative \(f'(x)=3x^{2}-3\) is a quadratic function and its graph will be a parabolic curve.
3Step 3: Interpret the x-intercept
The x-intercept of the derivative, that is, the value of \(x\) when \(f'(x) = 0\), is an indicator of the stationary points (where the curve reaches a peak or trough) of the original function. To find the x-intercept, set \(f'((x)\) to 0 and solve for \(x\). Solving \(3x^{2}-3 = 0\), we find \(x = ±1\). These points are the stationary points of the original function. On the graph of \(f(x)=x^{3}-3x\), these points represent local minima and maxima.
Key Concepts
DerivativePower RuleGraphing FunctionsStationary Points
Derivative
A derivative helps us understand how a function changes at any given point. It's like measuring the slope or steepness of the graph at a particular point on a curve. In our exercise, we were tasked with finding the derivative of the function \( f(x) = x^3 - 3x \). This derivative, denoted as \( f'(x) \), helps to reveal the behavior of the function itself. By using calculus, we realize that the derivative of \( x^3 \) is \( 3x^2 \) and the derivative of \( -3x \) is \(-3\). Combine these to get \( f'(x) = 3x^2 - 3 \).
Understanding derivatives is crucial because they provide insights into the rate of change, which is applicable in various fields such as physics, engineering, and even economics.
Understanding derivatives is crucial because they provide insights into the rate of change, which is applicable in various fields such as physics, engineering, and even economics.
Power Rule
The power rule is a quick shortcut to finding derivatives, especially when dealing with polynomial functions. It states that if you have a function \( x^n \), its derivative is \( nx^{n-1} \). This means you multiply the exponent by the coefficient and decrease the exponent by one.
In our function \( f(x) = x^3 - 3x \), we apply the power rule to each term:
In our function \( f(x) = x^3 - 3x \), we apply the power rule to each term:
- For \( x^3 \), the derivative is \( 3x^2 \).
- For \( -3x \), since it's really \( -3x^1 \), the derivative is \( -3 \times 1 \times x^0 = -3 \).
Graphing Functions
Graphing functions gives us a visual representation of mathematical relationships. It allows us to see the "big picture" by showing how the function behaves over a range of values. In this exercise, we graphed both \( f(x) = x^3 - 3x \) and its derivative \( f'(x) = 3x^2 - 3 \) to observe their shapes and intersections.
The original function, \( f(x) \), is a cubic function resulting in a curve that can have changes in direction. The derivative \( f'(x) \) is a quadratic function, which creates a parabolic shape. By overlaying these graphs, we can see points of interest, such as where the derivative intersects the x-axis indicating stationary points on the original graph.
The original function, \( f(x) \), is a cubic function resulting in a curve that can have changes in direction. The derivative \( f'(x) \) is a quadratic function, which creates a parabolic shape. By overlaying these graphs, we can see points of interest, such as where the derivative intersects the x-axis indicating stationary points on the original graph.
Stationary Points
Stationary points occur where the derivative of a function equals zero. These points are significant because they reveal where the function's graph reaches its local maxima or minima, meaning the highest or lowest points in a particular section.
To find these stationary points for the function \( f(x) = x^3 - 3x \), we set the derivative \( f'(x) = 3x^2 - 3 \) to zero: \[ 3x^2 - 3 = 0 \] Simplifying this, we get: \[ x^2 = 1 \] Thus, \( x = \pm 1 \).
These \( x \)-values correspond to the peaks and troughs on the graph of \( f(x) \). Specifically, \( x = 1 \) and \( x = -1 \) are where the function pauses its increase or decrease momentarily, pointing out possible local highs or lows. Identifying these points is essential for analyzing the overall behavior of a function.
To find these stationary points for the function \( f(x) = x^3 - 3x \), we set the derivative \( f'(x) = 3x^2 - 3 \) to zero: \[ 3x^2 - 3 = 0 \] Simplifying this, we get: \[ x^2 = 1 \] Thus, \( x = \pm 1 \).
These \( x \)-values correspond to the peaks and troughs on the graph of \( f(x) \). Specifically, \( x = 1 \) and \( x = -1 \) are where the function pauses its increase or decrease momentarily, pointing out possible local highs or lows. Identifying these points is essential for analyzing the overall behavior of a function.
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