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Chapter 3: Problem 3

Determine the absolute extrema of each function on the given interval. Illustrate your results by sketching the graph of each function. a. \(f(x)=x^{2}-4 x+3,0 \leq x \leq 3\) b. \(f(x)=(x-2)^{2}, 0 \leq x \leq 2\) c. \(f(x)=x^{3}-3 x^{2},-1 \leq x \leq 3\) d. \(f(x)=x^{3}-3 x^{2}, x \in[-2,1]\) e. \(f(x)=2 x^{3}-3 x^{2}-12 x+1, x \in[-2,0]\) \(f(x)=\frac{1}{3} x^{3}-\frac{5}{2} x^{2}+6 x, x \in[0,4]\)

Short Answer

Expert verified
Evaluate each function at its critical points and endpoints within the intervals to find the absolute extrema.

Step by step solution

01

Define Critical Points

To find the absolute extrema of a function within an interval, we must first identify the critical points where the function's derivative is zero or undefined.
02

Calculate Derivative for Each Function

Compute the derivative for each function to determine where the critical points occur. This means differentiating the function with respect to x.
03

Solve for Critical Points

Solve the equation set by the derivative to identify critical points for each function within the specified interval.
04

Evaluate Function at Critical Points and Endpoints

Compute the function value at each critical point and at the endpoints of the interval to identify where the absolute maximum and minimum values occur.
05

Analyze Function and Graph for Extrema

By evaluating each critical point and endpoint, we can determine the absolute extrema. Sketch the graph if required to illustrate these differences.

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Key Concepts

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

Absolute Extrema
Finding the absolute extrema of a function within a specific interval involves determining the highest and lowest points the function reaches in that range. These values are vital because they show us the greatest and smallest outputs of the function:
  • The absolute maximum is the highest point on a graph, representing the largest value of the function in the interval.
  • The absolute minimum is the opposite, showing the smallest value the function attains in the given interval.
A function can have an absolute maximum or minimum at endpoints of the interval, or at any critical points within the interval. Thus, it's important to evaluate both the critical points and the endpoints to find these extrema. This ensures that all potential highest and lowest points are considered.
Critical Points
Critical points are pivotal in determining where a function's slope changes direction, either from increasing to decreasing or vice versa. We find these points by taking the derivative of the function, which is a mathematical expression that describes the function's rate of change.

To locate critical points:
  • Calculate the derivative of the function.
  • Set the derivative equal to zero to find where the slope is horizontal.
  • Identify any points where the derivative does not exist.
These steps allow us to determine where the function could have a relative maximum or minimum, essential in the broader process of finding absolute extrema within an interval.
Function Analysis
Analyzing a function involves understanding its overall behavior. This means not just finding its critical points, but also its intercepts, increasing/decreasing intervals, concavity, and asymptotic behavior, if applicable.
  • Intercepts: Points where the graph touches or crosses the axes.
  • Increasing/Decreasing Intervals: When the function's derivative is positive, the function increases; when negative, it decreases.
  • Concavity: Determines if the function curves upward or downward, found through the second derivative.
A good function analysis gives us a comprehensive view of the function's behavior in the desirable interval, making it easier to predict and understand the location of maxima and minima.
Graph Sketching
Sketching the graph of a function based on its analytical results is a crucial step in calculus. A visual representation aids in comprehending the function's behavior, portraying a clear picture of where the extrema lie within the interval.
To sketch a graph:
  • Evaluate critical points and endpoints, plotting them to establish key positions.
  • Consider any points of inflection, where concavity shifts.
  • Use intercepts and asymptotes to guide the curve accurately across the relevant sections.
By connecting these points and features smoothly, you'll create a graph that reflects the function's nature. This step not only reinforces earlier analytical steps but also visually confirms the locations of absolute extrema.

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Most popular questions from this chapter

A real estate office manages 50 apartments in a downtown building. When the rent is \(\$ 900\) per month, all the units are occupied. For every \(\$ 25\) increase in rent, one unit becomes vacant. On average, all units require \(\$ 75\) in maintenance and repairs each month. How much rent should the real estate office charge to maximize profits?

When an object is launched vertically from ground level with an initial velocity of \(40 \mathrm{m} / \mathrm{s}\), its position after \(t\) seconds is \(s(t)=40 t-5 t^{2}\) metres above ground level. a. When does the object stop rising? b. What is its maximum height?

Sandy is making a closed rectangular jewellery box with a square base from two different woods. The wood for the top and bottom costs \(\$ 20 / \mathrm{m}^{2}\). The wood for the sides costs \(\$ 30 / \mathrm{m}^{2}\). Find the dimensions that will minimize the cost of the wood for a volume of \(4000 \mathrm{cm}^{3}.\)

The polynomial function \(f(x)=0.001 x^{3}-0.12 x^{2}+3.6 x+10,0 \leq x \leq 75\) models the shape of a roller-coaster track, where \(f\) is the vertical displacement of the track and \(x\) is the horizontal displacement of the track. Both displacements are in metres. Determine the absolute maximum and minimum heights along this stretch of track.

A rectangular piece of land is to be fenced using two kinds of fencing. Two opposite sides will be fenced using standard fencing that costs \(\$ 6 / \mathrm{m},\) while the other two sides will require heavy-duty fencing that costs \(\$ 9 / \mathrm{m}\). What are the dimensions of the rectangular lot of greatest area that can be fenced for a cost of \(\$ 9000 ?\)
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