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Chapter 4: Q. 67 (page 242)

For what positive numbers will the cube of a number exceed four times its square?

Short Answer

Expert verified

The cube of a number will exceed four times its square for every positive value of x in the set of{x|x>4}orintervalnotation(4,鈭�).

Step by step solution

01

Step 1. Given Information

We have to find out for what positive numbers will the cube of a number exceed four times its square.

02

Step 2. Form an inequality expression

To find the positive numbers, let the number be x.

According to the question,

x3>4x2x3-4x2>0

03

Step 3. Solve the inequality expression

Let f(x)=x3-4x2

Now, let's find the real zeros

x3-4x2=0x2x-4=0x2=0andx-4=0x=0x=4

04

Step 4. Check the points

From the real zeros, we get three intervals which are -鈭�,0,0,4,4,鈭�.

Let x=-2, substitute in fx=x3-4x2

f-2=-23-4-22f-2=-8-16f-2=-24

It is negative.

Let x=2, substitute in fx=x3-4x2

f2=23-422f2=8-16f2=-8

It is negative.

Let x=6, substitute in fx=x3-4x2

f6=63-462f6=72

It is positive.

Thus, the set is {x|x>4}orintervalnotation4,鈭�where the cube of a number will exceed four times its square.

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