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Chapter 3: Q. 32 (page 261)

Use a sign chart for f'to determine the intervals on which each function fis increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

f(x)=ex1+ex

Short Answer

Expert verified

Ans: It has no solution hence this is a constant function.

Intervals of the given function are

Increasing at (-3,鈭�)

Step by step solution

01

Step 1. Given information:

f(x)=ex1+ex

02

Step 2. Finding the derivative of the function:

f(x)=ex1+exf'(x)=(1+ex)ex-ex(ex)(1+ex)2鈭�鈭�xuv=v鈭�鈭�x(u)-u鈭�鈭�x(v)v2f'(x)=ex+e2x-e2x(1+ex)2=ex(1+ex)2f'(x)=ex(1+ex)2letf'(x)=0鈭�ex(1+ex)2=0ex=(1+ex)2脳0ex=0ex=0x=nosolutions

03

Step 3. Inserting the root points on the number line(Sign chart):

It has no solution hence this is a constant function.

so according to the graph, we can say that the function increases from 1

Therefore(-3,鈭�)

04

Step 4. Verifying algebraic answers with graphs :

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Most popular questions from this chapter

For each graph of f in Exercises 49鈥�52, explain why f satisfies the hypotheses of the Mean Value Theorem on the given interval [a, b] and approximate any values c 鈭� (a, b) that satisfy the conclusion of the Mean Value Theorem.

For each set of sign charts in Exercises 53鈥�62, sketch a possible graph of f.

Determine whether or not each function f in Exercises 53鈥�60 satisfies the hypotheses of the Mean Value Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c 鈭� (a, b) that satisfy the conclusion of the Mean Value Theorem.

f(x)=x2鈭�1x2鈭�4,[a,b]=[鈭�3,3]

Find the possibility graph of its derivative f'.

For each set of sign charts in Exercises 53鈥�62, sketch a possible graph of f.
See all solutions

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