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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 2, Problem 8P

**(a)**

To determine

**To Sketch:** A continuous function with domain [0,1] and range is also lies in [0,1]. It has fixed point.

Expert Solution

The graph of the curve is *p.* The function *f* is continuous at *p.*

The function *f* continuous with domain [0,1] and range is also lies in [0,1].

**(b)**

To determine

**To Draw:** A continuous function with domain [0,1] and range is also lies in [0,1]. It does not have fixed point.

Expert Solution

There is no possible to draw the graph of the continuous function with domain [0,1] and range is also lies in [0,1]. It does not have fixed point.

Because, the “obstacle” is the line *p.* If the graph doesn’t cross the line somewhere in

(c)

To determine

**To prove:** Any continuous function with domain [0,1] and range in [0,1] must have a fixed point.

Expert Solution

**Result used:** The intermediate value theorem,

If *w* between *c* in between *a* and *b* such that

**Proof:**

Consider the function

Now need prove *f* has a fixed point that means if *f* has a fixed point.

Assume that

and

By the intermediate value theorem, there exist some number *c* in the interval

So

Therefore *f* has a fixed point *c.*

Hence proved.