The function (x) = |x| has an absolute minimum value at x = 0 even though
Question:
The function ƒ(x) = |x| has an absolute minimum value at x = 0 even though ƒ is not differentiable at x = 0. Is this consistent with Theorem 2? Give reasons for your answer.
Theorem 2
Transcribed Image Text:
THEOREM 2-The First Derivative Theorem for Local Extreme Values If f has a local maximum or minimum value at an interior point c of its domain, and if f' is defined at c, then Proof To prove that f'(c) is zero at a local extremum, we show first that f'(c) cannot be positive and second that f'(c) cannot be negative. The only number that is neither posi- tive nor negative is zero, so that is what f'(c) must be. To begin, suppose that f has a local maximum value at x = c (Figure 4.6) so that f(x) = f(c) ≤ 0 for all values of x near enough to c. Since c is an interior point of f's domain, f'(c) is defined by the two-sided limit Similarly, f'(c) = lim This means that the right-hand and left-hand limits both exist at x = c and equal f'(c). When we examine these limits separately, we find that Because (x-c) > 0 and f(x) = f(c) f'(c) = lim X-C f'(c) = 0. f(x) = f(c) x c f(x) = f(c) - x-c f(x) = f(c) x - C lim ≤ 0. ≥ 0. Because (x c) < 0 and f(x) = f(c) (1) (2)
Fantastic news! We've Found the answer you've been seeking!
Step by Step Answer:
Answer rating: 66% (9 reviews)
ANSWER The given function x x has an absolute minimum value at x 0 even though it is not differentia...View the full answer
Answered By
Brian Kiprono
Taking part in public speeches, blogging, writing essays
0.00
0 Reviews
10+ Question Solved
Related Book For
Thomas Calculus Early Transcendentals
ISBN: 9780321884077
13th Edition
Authors: Joel R Hass, Christopher E Heil, Maurice D Weir
Question Posted:
Students also viewed these Mathematics questions
-
a. Give an example of a differentiable function whose first derivative is zero at some point c even though has neither a local maximum nor a local minimum at c. b. How is this consistent with...
-
Q. The Executive Chairman of Zoomlion and Jospong Groups of Companies, Dr. Joseph Siaw Agyepong has built up one of the Africas largest sanitation management businesses, becoming Ghanas wealthiest...
-
Which of the functions satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers. is zero at x = 0 and x = 1 and differentiable on (0,...
-
1. Resolve Class C 192.168.23.36 /27 2. Design DMZ minimumof 4 servers assigning an IP addressfor all devices 3. List and explain 4 primary servers that will be in the DMZ 4. Resolve Class C 192.168...
-
Explain the critical differences between basic service, customer satisfaction, and customer success.
-
Explain how it is possible for an entity with a high current ratio to have difficulty paying its bills.
-
Describe how you think history is repeating itself in todays healthcare system.
-
Jane Myers is the manager of an extremely successful gift shop, Janes Gifts, which is operated for the benefit of local charities. From the data below, she wants a cash budget showing expected cash...
-
. An urn has 10 balls that are identical except that7are white and 3 are red. A sample of 8 is selected randomly without replacement. What is the probability that exactly 6 are white and2are red?...
-
Music-Is-Us, Inc., is a supplier of musical instruments for professional and amateur musicians. The companys accountants make adjusting entries monthly, and they make all closing entries annually....
-
Find a value of c that makes the function continuous at x = 0. Explain why your value of c works. f(x) = 9x C, 3 sin 3x 5x x = 0 x = 0
-
Find the absolute maximum value of (x) = x 2 ln (1/x) and say where it is assumed.
-
Calculate and print out the balance due, principal payment, and interest payment for each period of a used-car loan. The nominal interest is 12% per year, compounded monthly. Payments are made...
-
stimulus go venture CEO, I selling electric car compete friends in class. 1. What is your innovation strategy for your firm in the simulation, drawing on the readings/videos of this week? 2. How were...
-
Select a business that uses e-commerce and social commerce. Research some of the security threats and challenges it faces and how it overcomes them. write short introduction, background, Research...
-
For the Case Study, AstraZeneca: Fighting a global pandemic (2021). Describe the nature of the project. Analyze the project scope and importance of managing scope change. Assess the importance of...
-
1. Prepare a table explaining how you would apply the performance domain with respect to the stakeholder engagement aspects of the Real Milk company?: INVOLVEMENT OF INTERESTED PARTIES EXPLANATION...
-
A strategic issue is a fundamental policy choice , or change challenge , affecting Walmart Inc mandates, mission, product or service level and mix, clients or users, costs, financing, organization,...
-
Why do accountants and economists calculate a firms cost and profit in different ways?
-
Provide a few individual examples who revealed what aspects of emotional intelligence?
-
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the...
-
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the...
-
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the...
-
Basic CVP Concepts Becca's Bakery sells cupcakes at $3.15 apiece. The variable cost per cupcake is $1.10. The fixed costs are $12,500 per month. Calculate the unit contribution margin if unit...
-
Prove that the formula for the critical buckling load of a column with a total height (L) when one end is fixed, and the other end is pinned is given by: Per = 2.045PE where PE is the Euler buckling...
-
Look at the slanted line BC, which represents the limit of one of the constraints and runs between the corners B and C. For specificity let point B = (bx,by) and let C = (Cx,Cy). Suppose that the...
Study smarter with the SolutionInn App