Problem 97 - Calculus Single Variable

Question

In each of Exercises \(91-94\), calculate and plot the derivative \(f^{\prime}\) of the given function \(f\). Use this plot to identify candidates for the local extrema of \(f\). Add the plot of \(f\) to the window containing the graph of \(f^{\prime} .\) From this second plot, determine the behavior of \(f\) at each candidate for a local extremum. $$ f(x)=x^{4}-2 x^{3}+x^{2}-2 x+13, \quad I=[-1,3] $$

Step-by-Step Solution

Answer

The potential local extrema occur at the zeros of the derivative on [-1, 3]. Analyze the changes in sign of the derivative to classify them.

Step 1: Find the derivative of the function

First, we need to calculate the derivative of the function \( f(x) = x^4 - 2x^3 + x^2 - 2x + 13 \). We do this by applying standard differentiation rules:\[ f'(x) = \frac{d}{dx}(x^4) - \frac{d}{dx}(2x^3) + \frac{d}{dx}(x^2) - \frac{d}{dx}(2x) + \frac{d}{dx}(13) \]Calculating each term gives us:\[ f'(x) = 4x^3 - 6x^2 + 2x - 2 \].

Step 2: Plot the derivative

Using a graphing tool or software, plot the derivative \( f'(x) = 4x^3 - 6x^2 + 2x - 2 \) over the interval \([-1, 3]\). The plot will help us identify where the function changes direction, suggesting possible local extrema (maxima or minima). Look for roots of \( f'(x) \) where it crosses the x-axis.

Step 3: Identify candidates for local extrema

From the plot of \( f'(x) \), identify the x-values (in the interval \([-1, 3]\)) where the derivative is zero or changes from positive to negative (indicating a local maximum) or negative to positive (indicating a local minimum). Solve \( 4x^3 - 6x^2 + 2x - 2 = 0 \) for these values.

Step 4: Plot the original function

Now plot the original function \( f(x) = x^4 - 2x^3 + x^2 - 2x + 13 \) on the same set of axes as \( f'(x) \). This helps to visualize the behavior of \( f(x) \) at the candidates for local extrema identified earlier.

Step 5: Analyze the behavior at each critical point

Examine the plot of \( f(x) \) at the x-values where \( f'(x) = 0 \) or changes sign. If the function \( f(x) \) increases to the left and decreases to the right of a point, this point is a local maximum. Conversely, if it decreases to the left and increases to the right, it's a local minimum. If it doesn't change, then it's neither.

Key Concepts

Derivative

In calculus, the derivative of a function gives us essential information about the function's rate of change. The derivative can be thought of as a formula that tells us the slope of the tangent line at any point on a function's graph. In our exercise, the function we are dealing with is a polynomial:
\[ f(x) = x^4 - 2x^3 + x^2 - 2x + 13 \]
To find its derivative, we apply polynomial differentiation rules, which in basic terms, involve using the power rule for each term. The derivative of the function is:\[ f'(x) = 4x^3 - 6x^2 + 2x - 2 \]
By finding this derivative, we can analyze the behavior of the function, particularly looking into how it increases or decreases as we move along the x-axis. This understanding is crucial when we later analyze critical points and local extrema.

Local Extrema

Local extrema refer to the highest or lowest points (maximums or minimums) within a given region of a function. These are points where the function can change direction from increasing to decreasing or vice versa.
To identify these points, we look at the derivative. Specifically, local extrema occur at critical points where the derivative is zero or undefined, and the function changes its behavior at these points.

• A **local maximum** is where the derivative changes from positive to negative.
• A **local minimum** is where it changes from negative to positive.

In our exercise, after finding the derivative \( f'(x) = 4x^3 - 6x^2 + 2x - 2 \), we look for values of \( x \) in the interval \( [-1, 3] \) where this derivative equals zero or changes sign. Solving \( 4x^3 - 6x^2 + 2x - 2 = 0 \) will give these critical points.

Graphing Functions

Graphing functions is a visual way to understand the behavior of functions over a particular interval. By graphing a function alongside its derivative, we gain insights into how the function behaves in terms of increases and decreases within certain intervals.
To graph the original function \( f(x) = x^4 - 2x^3 + x^2 - 2x + 13 \) and its derivative \( f'(x) \), we use graphing software or tools.

• The graph of the derivative helps identify critical points, which signal potential extrema.
• The graph of the original function helps us confirm the behavior at these points.

By overlaying both graphs, we can visually connect changes in the derivative to changes in the function. This comprehensive picture is crucial to identifying and validating local maxima and minima.

Critical Points

Critical points are where the first derivative of a function is either zero or undefined. These points are important because they are potential locations for local extrema - the peaks and troughs of the function's graph.
To find critical points, we solve the derivative equation \( f'(x) = 4x^3 - 6x^2 + 2x - 2 = 0 \).

• When \( f'(x) = 0 \), the slope of the tangent to the curve is zero, indicating a potential local maximum or minimum.
• I f \( f'(x) \) changes from positive to negative at a point, that point is a local maximum.
• If it changes from negative to positive, it's a local minimum.

Once critical points are located, graphing the original function and observing its behavior around these points helps confirm whether they are indeed extrema and what type they represent.