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Chapter 3: Q. 40 (page 261)

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.

f(x)=x2(x-1)(x+1)

Short Answer

Expert verified

Ans: The local extrema value of f(x) are

Max at x=0

Min at x=卤12

Step by step solution

01

Step 1. Given Information

f(x)=x2(x-1)(x+1)

02

Step 2. Finding the derivative of the function

Rewriting the function by simplifying it,

f(x)=x2(x-1)(x+1)=x2(x2-1)=x4-x2f'(x)=4x3-2xlet,f'(x)=04x3-2x=02x(2x2-1)=02x=0x=02x2-1=02x2=1x=卤12x=0,卤12

03

Step 3. Substituting the values into the function equation:

x=-2,12,2intothef'(x)=x4-x2f'(-2)=(-2)4-(-2)2=12>0f'(2)=(2)4-(2)2=12>0f'(12)=(12)4-(12)2=-316<0

04

Step 4. Finding local extrema of the function on the number line(Sign chart): 

05

Step 5. Verifying algebraic answers with graphs :

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Most popular questions from this chapter

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)=2x,[a,b]=[0,3]

Find the critical points of the function

f'(x)=(x-1)(x-2)x-3

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)

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

It took Alina half an hour to drive to the grocery store that is 20 miles from her house.

(a) Use the Mean Value Theorem to show that, at some point during her trip, Alina must have been traveling exactly 40 miles per hour.

(b) Why does what you have shown in part (a) make sense in real-world terms?
See all solutions

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