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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.1.9
Chapter 4, Problem 4.1.9

Finding Extrema from Graphs


In Exercises 7–10, find the absolute extreme values and where they occur.


Graph showing a line segment from point (2, 0) to point (2, 5) on a coordinate plane, indicating extrema.

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Identify the endpoints of the graph. The graph starts at the point (0, 5) and ends at the point (2, 0).
Determine the type of extrema by analyzing the graph. The graph is a straight line decreasing from (0, 5) to (2, 0).
The absolute maximum value occurs at the highest point on the graph, which is at (0, 5).
The absolute minimum value occurs at the lowest point on the graph, which is at (2, 0).
Verify that there are no other points on the graph that exceed the y-values at the endpoints, confirming the absolute extrema.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute Extrema

Absolute extrema refer to the highest or lowest points on a graph over a given interval. The absolute maximum is the highest point, while the absolute minimum is the lowest. These points can occur at critical points or endpoints of the interval. In the context of the graph, identifying these points involves examining the y-values at the endpoints and any critical points within the interval.
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Finding Extrema Graphically

Endpoints of a Graph

Endpoints are the points at the boundaries of a graph's domain. They are crucial when determining absolute extrema, as extrema can occur at these points. In the given graph, the endpoints are at (2, 5) and (2, 0). Evaluating the function's value at these points helps in identifying the absolute maximum and minimum values.
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Critical Points

Critical points are where the derivative of a function is zero or undefined, indicating potential local maxima or minima. However, in a linear segment like the one shown, there are no critical points within the interval, as the slope is constant. Thus, for this graph, the focus is on the endpoints to find the absolute extrema.
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Related Practice
Textbook Question

Business and Economics

62. Production level Suppose that c(x)=x^3-20x^2 + 20,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items.

Textbook Question

A cone is formed from a circular piece of material of radius 1 meter by removing a section of angle θ and then joining the two straight edges. Determine the largest possible volume for the cone.

Textbook Question

Identifying Extrema


In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.


Textbook Question

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.

2. y=x^4/4-2x^2+4

Textbook Question

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

77. y' = cot(θ/2), for 0 < θ < 2π

Textbook Question

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(1 + cos 4t)/2 dt