Differentiability on Real Numbers (R)

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Introduction to Analysis › Differentiability on Real Numbers (R)

Questions 1 - 2

1

Determine whether the following statement is true or false:

Let , , , and $d\ \epsilon\ \mathbb{R}$ . If and then .

False

True

Explanation

Determine this statement is false by showing a contradiction when actual values are used.

Let

First make sure the inequalities hold true.

$\a<b \1<2$

and

$\c<d<0 \-3<-2<0$

Now find the products.

$\ac<bd \(1)(-3)<(2)(-2) \-3 less -4$

Therefore, the statement is false.

2

Determine whether the following statement is true or false:

Let , , , and $d\ \epsilon\ \mathbb{R}$ . If and then .

False

True

Explanation

Determine this statement is false by showing a contradiction when actual values are used.

Let

First make sure the inequalities hold true.

$\a<b \1<2$

and

$\c<d<0 \-3<-2<0$

Now find the products.

$\ac<bd \(1)(-3)<(2)(-2) \-3 less -4$

Therefore, the statement is false.

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