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Chapter 4: Problem 54

The derivative \(f^{\prime}\) of a function \(f\) is given. Use u graphing utility to graph \(f^{\prime}\) on the indicated interval. Estimate the critical points of \(f\) and determine al each such point whether \(f\) has a local maximum, a local minimum, or nether. Round off your estimates to three decimal places. $$f^{\prime}(x)=2 x^{3}+x^{2}-4 x+3 ;[-4,4]$$.

Short Answer

Expert verified
The exact answer will depend on the graph plotted, but the procedure will be to identify the points where the derivative crosses the x-axis. Then for each of these points, determine if \(f^{\prime}(x)\) changes from positive to negative (local max), from negative to positive (local min), or stays the same (neither) as x increases.

Step by step solution

01

Graph the Derivative

Using a graphing utility plot the function \(f^{\prime}(x)=2 x^{3}+x^{2}-4 x+3\) on the interval \([-4,4]\). Keep in mind that critical points occur where the function is zero or undefined. These are typically the points where the graph intersects the x-axis.
02

Identify Critical Points

From the graph of \(f^{\prime}\), identify and estimate the x-coordinates of the points where the graph crosses the x-axis. These x-coordinates denote the critical points of the original function \(f(x)\). Round off your estimates to three decimal places.
03

Determine Whether Each Critical Point is a Local Maximum, Local Minimum, or Neither

Look at the graph and note the behavior of the derivative on either side of the critical points. If the derivative changes its sign from positive to negative as x increases through a critical point, then the function \(f\) has a local maximum at that critical point. If the derivative changes sign from negative to positive, then the function \(f\) has a local minimum. If there is no sign change in the derivative through a critical point, then \(f\) doesn't have a local extremum (maximum or minimum) at that point.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative Analysis
The concept of derivative analysis is vital in understanding the behavior of a function. A derivative, represented by \(f'(x)\), provides the rate of change of the function \(f(x)\) at any given point. It essentially tells us how steep the graph of \(f(x)\) is at a particular point.
  • If \(f'(x) > 0\), it indicates that \(f(x)\) is increasing at that point.
  • If \(f'(x) < 0\), \(f(x)\) is decreasing.
  • When \(f'(x) = 0\), \(f(x)\) may change from increasing to decreasing or vice versa.
In the exercise, the derivative given is a cubic polynomial \(f'(x) = 2x^3 + x^2 - 4x + 3\). Cubic functions can have up to three real roots, meaning that there can be up to three critical points where the derivative is zero.
Critical Points
Critical points occur where the derivative of a function is zero or undefined. They are essential in calculus because they can indicate where the function has local maxima, minima, or points of inflection. In our exercise, we are tasked with identifying these points by using a graphing utility.
  • Plot the derivative \(f'(x)\) on the given interval \([-4, 4]\).
  • Find where the graph intersects the x-axis, as these intersections indicate where \(f'(x) = 0\).
Estimation of these x-values should be accurate to three decimal places. These estimates give us the critical points for the original function \(f(x)\). Critical points are crucial in further analysis to determine if they represent local maxima, minima, or neither.
Local Maximum and Minimum
Local maxima and minima are points where a function reaches a highest or lowest value locally, meaning in a small neighborhood around those points. To determine if a critical point is a local maximum or minimum, observe the sign change of the derivative around these points:
  • If \(f'(x)\) changes from positive to negative at a critical point, \(f(x)\) has a local maximum there.
  • If \(f'(x)\) changes from negative to positive, the function has a local minimum.
  • No sign change means the point could be an inflection point, and is neither a maximum nor a minimum.
This step is essential in understanding how \(f(x)\) behaves around those critical points and is useful in sketching the function's graph.
Graphing Utilities
Graphing utilities are powerful tools in calculus that help visualize functions and their derivatives. They provide visual insights into the behavior of a function that may not be clear through algebra alone.To use a graphing utility effectively:
  • Enter the derivative function \(f'(x) = 2x^3 + x^2 - 4x + 3\) in the tool.
  • Set the x-axis to the interval \([-4, 4]\) to focus on the relevant section.
  • Observe where the derivative graph crosses the x-axis to find critical points.
Graphing makes it easier to estimate critical points accurately and assess whether they are local maxima or minima. It also provides a clearer understanding of how the derivative influences the behavior of the original function.

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Most popular questions from this chapter

A man star's at a point \(A\) and walks 40 feet north. He then turns and walks due cast at 4 feet per second. A scarchlight placed at \(A\) follows him. Al what rate is the light turning 15 seconds after the man started walking cast?

(Simple harmonic motion) A bob suspended from a spring oscillates up and down about an equilibrium point, its vertical position at time \(t\) given by $$y(t)=A \sin \left(\omega t+\varphi_{0}\right)$$ where \(A, \omega, \varphi_{0}\) are positive constants. (This is an idealization in which we are disregarding friction.) (a) Show that at all times \(t\) the acceleration of the bob \(y^{\prime \prime}(l)\) is related to the position of the bob by the equation $$y^{\prime \prime}(t)+\omega^{2} y(t)=0$$. (b) It is clear that the bob oscillates from \(-A\) to \(A,\) and the speed of the bob is zero at these points. At what position does the bob attain maximum speed? What is this maximum speed? (c) What are the extreme values of the acceleration function? Where does the bob attain these extreme values?

The function \(f(x)=x^{4}-2 x^{2}-\frac{17}{16}\) has two zeros, one at a point \(a\) between \(0\) and \(2,\) and the other at \(-a . C\) is an even function.) (a) Show that the Newton-Raphson method fails in the search for \(a\) if we start at \(x=\frac{1}{2} .\) What are the outputs \(x_{1}, x_{2}, x_{3}, \ldots\) in this case? (b) Estimate \(a\) by starting at \(x_{1}=2 .\) Determine \(x_{4}\) rounded off to five decimal places and evaluate \(f\left(x_{4}\right)\)

Neglect air resistance. For the numerical calculations, take \(g\) as 32 feet per second \(\mathrm{p}\) er second or as 9.8 meters per second per second. An object projected vertically upward passes every height less than the maximum twice, once on the way up and once on the way down. Show that the speed is the same in each direction. Measure height in feet.

Sketch the graph of the function using the approach presented in this section. $$f(x)=\sqrt{\frac{x}{x+4}}$$
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