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Chapter 3: Q. 12 (page 260)

Prove that the function is increasing on all values of real numbers.

f(x)=x3.

Short Answer

Expert verified

Thefunctionf(x)=x5isincreasingonallofrealnumber.

Step by step solution

01

Step 1. Given 

The given function isf(x)=x3.

02

Step 2.  Finding the intervals

Now,ifb<a≤0thenb5<a5Thus,thefunctionf(x)=x5isincreasingintheinterval(-∞,0].Now,if0<a<bthena5<b5Thus,thefunctionf(x)=x5isincreasingintheinterval[0,∞).Therefore,thefunctionf(x)=x5isincreasingintheinterval(-∞,∞).

03

Step 3. Prove

Thederivativeofthefunctionisgivenbyf'(x)=5x4Whichisalwayspositiveforx∈R.Therefore,thefunctionf(x)=x5isincreasingonallofrealnumber.

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Most popular questions from this chapter

Find the critical points of the function

f'(x)=1+x2-4

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)=cos2(x)

Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of f,f',andf'', and examine any relevant limits so that you can describe all key points and behaviors of f.

f(x)=1x-12x-2

Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of f,f',andf'', and examine any relevant limits so that you can describe all key points and behaviors of f.

f(x)=x3(x+2)

Determine whether or not each function f in Exercises 53–60 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 ∈ (a, b) that satisfy the conclusion of the Mean Value Theorem.

f(x)=−x3+3x2−7,[a,b]=[−2,3]
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