Problem 43 - Calculus A New Horizon

Question

Use a graphing utility to generate the graphs of \(f^{\prime}\) and \(f^{\prime \prime}\) over the stated interval, and then use those graphs to estimate the \(x\) -coordinates of the relative extrema of \(f .\) Check that your estimates are consistent with the graph of \(f\). f(x)=x^{4}-24 x^{2}+12 x, \quad-5 \leq x \leq 5

Step-by-Step Solution

Answer

Graph \( f'(x) = 4x^3 - 48x + 12 \) and \( f''(x) = 12x^2 - 48 \) to find critical points and use second derivative to confirm. Verify with \( f(x) \) graph.

Step 1: Find the First Derivative

To find the first derivative of the given function, apply the power rule to each term of the function. The function is \( f(x) = x^4 - 24x^2 + 12x \), therefore the first derivative is:\[ f'(x) = 4x^3 - 48x + 12 \].

Step 2: Find the Second Derivative

Next, we need to find the second derivative to identify the concavity and points of inflection which help in determining the extremum points. Differentiate \( f'(x) = 4x^3 - 48x + 12 \) to get:\[ f''(x) = 12x^2 - 48 \].

Step 3: Graph the First Derivative

Using a graphing utility, plot the graph of the first derivative \( f'(x) = 4x^3 - 48x + 12 \) over the interval \(-5 \leq x \leq 5\). Look for the values of \( x \) where the graph crosses the x-axis, as these points indicate potential relative extrema (where \( f'(x) = 0 \)).

Step 4: Graph the Second Derivative

Next, graph the second derivative \( f''(x) = 12x^2 - 48 \) over the same interval. Check where the graph crosses or touches the x-axis, which shows the points of inflection or changes in concavity. This helps determine whether each critical point found is a maximum or a minimum.

Step 5: Estimate Critical Points and Extremas

From the graph of the first derivative, estimate the x-coordinates of the critical points where \( f'(x) = 0 \). Use the graph of \( f''(x) \) to decide if these points are maxima or minima. If the second derivative is positive at a critical point, it is a minimum; if negative, it is a maximum.

Step 6: Verify with the Graph of Original Function

Finally, graph \( f(x) = x^4 - 24x^2 + 12x \) over the interval \(-5 \leq x \leq 5\) to verify the estimations. Check if the behavior of the graph (increasing and decreasing) matches the analysis of the extrema from derivative graphs.

Key Concepts

Derivatives

In calculus, derivatives represent the rate at which a function changes at any given point. They are foundational to understanding the behavior of functions. Let's take a closer look at how to derive such expressions.

• The first derivative, often written as \( f'(x) \), reveals the slope or gradient of the original function \( f(x) \).
• The key to finding the derivative is applying the power rule. For a term like \( ax^n \), the derivative is \( nax^{n-1} \).
• In our exercise, \( f(x) = x^4 - 24x^2 + 12x \), so by applying the power rule, the derivative is \( f'(x) = 4x^3 - 48x + 12 \).

This expression helps us understand how the original function changes across its domain.

Concavity

Concavity describes the direction a graph bends. To determine this, we use the second derivative, \( f''(x) \). Here's how it works:

• If \( f''(x) > 0 \), the function is concave up, resembling a U-shape.
• If \( f''(x) < 0 \), the function is concave down, like an upside-down U or an N-shape.
• In our exercise, the second derivative \( f''(x) = 12x^2 - 48 \) informs us of the concavity.

These concavity tests also help determine inflection points where the graph changes from concave up to concave down or vice versa.

Relative Extrema

Relative extrema are the "peaks" and "valleys" found within the graph of a function. They indicate the local maximums or minimums points. Let's explore how to find and verify them:

• The zeros of the first derivative, \( f'(x) = 0 \), suggest where relative extrema may occur.
• Use the second derivative test:
  • If \( f''(x) > 0 \) at a critical point, that point is a local minimum.
  • If \( f''(x) < 0 \) at a critical point, it signifies a local maximum.
• By solving \( 4x^3 - 48x + 12 = 0 \), we find potential critical points and apply \( f''(x) \) at these points to classify them.

Achieving accurate results involves plotting these on the graph of the original function \( f(x) \) to visualize and confirm the extremas.

Graphing Utility

Graphing utilities are powerful tools in calculus, used to visualize and analyze functions comprehensively.

• Graphing tools can plot both the first and second derivatives, helping identify where \( f'(x) \) and \( f''(x) \) cross the x-axis.
• The intersection points aid in estimating the x-coordinates of potential relative extrema.
• Verification involves comparing plots of the original function, \( f(x) = x^4 - 24x^2 + 12x \), with its derivatives to see consistency in trends.

These visualizations not only support accurate calculation but also enhance understanding of function behaviors and properties.