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Get Full Access to Calculus: Early Transcendentals - 1 Edition - Chapter 4.1 - Problem 53e
Get Full Access to Calculus: Early Transcendentals - 1 Edition - Chapter 4.1 - Problem 53e

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# Absolute maxima and minima a. Find the | Ch 4.1 - 53E ISBN: 9780321570567 2

## Solution for problem 53E Chapter 4.1

Calculus: Early Transcendentals | 1st Edition

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Problem 53E

Absolute maxima and minima a. Find the critical points off on the given interval. b. Determine the absolute extreme values off on the given interval. c. Use a graphing utility to confirm your conclusions. 3 ?x f(x) = x e , on [-1,5]

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Solution 53E Step-1 Critical value definition; Let f be a continuous function defined on an open interval containing a number 1 1 ‘c’.The number ‘c’ is critical value ( or critical number ). If f (c) = 0 or f (c) is undefined. A critical point on that graph of f has the form (c,f(c)). Step-2 Absolute extreme value definition; When an output value of a function is a maximum or a minimum over the entire domain of the function, the value is called the absolute maximum or the absolute minimum. Let f be a fu nction with domain D and let c be a fixed constant in D . hen the output value f (c) is the 1. Absolute maximum value of f on D if and only if f(x) f(c) , for all x in D. 2. Absolute minimum value of f on D if and only if f(c) f(x) , for all x in D. Step_3 3 x a). Thegiven function is f(x) = x e on [-1,5] .Clearly the function is a exponential function and it is continuous for all of x . Now , we have to find out the critical points of f on the given interval. Now , f(x) = x e 3 x then differentiate the function both sides with respect to x. 3 dx f(x) = dx(x e x ) 1 3 d x x d 3 d d(v) d(u) f (x) = x dx (e )+ e dx (x ), since dx(uv) = u dx +v dx 3 2 =x ( (1) e x )+ e x/2 (3x ) 3 x x/2 2 3 x 2 x = (-x e )+ e (3x ) = (-x e )+ (3x e ) 2 = x e x ( -x +3) 1 2 c Since , from the definition f (c)=0 = c e ( - c+3) 2 c e c ( - c+3) = 0 c 2 That is , e =0 , c = 0 , and (-c+3) = 0 ln( e c ) =ln(0) , c = 0 , and c=3. (c)ln(e) = undefined , c = 0 , and c= 3. Therefore, c = 0 , 3. Clearly x = 0 , 3 lies between [-1,5]. 3 0 At x = 0, then f(0) = 0 e = 0 . 3 3 At x = 3, then f(3) = 3 e = 27 (0.0498025) = 1.34466 . Therefore ,f(x) attains the critical values at x = 0 3, and the critical points are (0,0) , (3, 1.24466). Step-4 b). Now , we have to determine the absolute extreme values of f on the given interval ; Here the given interval is [ -1 ,5] , and the function attains the critical value at x = 0, 3.So the endpoints are -1,5 From the step-2 , the absolute extreme values are; 3 Therefore , at x = 0, then f(0) = 0 e 0 = 0 . 3 At x= -1 , then f(-1) = (1) e (1) = - (e) = -2.718 3 3 At x = 3, then f(3) = 3 e = 27 (0.0498025) = 1.34466 , since, e = 2.718 3 5 At x = 5, then f(5) = 5 e = 125(0.00674144) = 0.842680, since, e = 2.718 Therefore , f(0) =0 , f(-1) = - 2.718 , f(3) = 1.34466 ,and f(5) = 0.842680 Hence , the largest value of f(x) is 1.34466 ,this attains at x= 3 Therefore , the absolute maximum is f( 3) = 1.34466 Hence , the smallest value of f(x) is -2.718,this attains at x = -1 Therefore , the absolute minimum is f( 1 ) = -2.718 Step_5 c) . The graph of the related function f(x) = x e Hence, from the above graph all the extreme values are true.

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##### ISBN: 9780321570567

The answer to “Absolute maxima and minima a. Find the critical points off on the given interval. b. Determine the absolute extreme values off on the given interval. c. Use a graphing utility to confirm your conclusions. 3 ?x f(x) = x e , on [-1,5]” is broken down into a number of easy to follow steps, and 43 words. This full solution covers the following key subjects: absolute, interval, given, graphing, determine. This expansive textbook survival guide covers 85 chapters, and 5218 solutions. Since the solution to 53E from 4.1 chapter was answered, more than 387 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 53E from chapter: 4.1 was answered by , our top Calculus solution expert on 03/03/17, 03:45PM. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321570567. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 1.

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