Math Is Fun Forum

Prakash Panneer

Member

Registered: 2006-06-01

Posts: 110

Please Please....

Give an example of a function on a closed interval which satisfies the conclusion of the Mean Value Theorem, but not the hypothesis.

Thanks in advance

---

Letter, number, arts and science
of living kinds, both are the eyes.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Please Please....

f(x) =

x when x does not equal 2
3 when x equals 2

interval [2, 3]

It's not smooth, and thus, does not satisfy the hypothesis.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Prakash Panneer

Member

Registered: 2006-06-01

Posts: 110

Re: Please Please....

Are you sure?

I think it doesn't satisfy the conclusion.

Right?

---

Letter, number, arts and science
of living kinds, both are the eyes.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Please Please....

Whoops, you are correct.  Make that:

f(x) =

x when x does not equal 2.1
3 when x equals 2.1

interval [2, 3]

So we need to find a point within [2,3] where the slope is 1.  This works for all values other than x = 2.1.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Prakash Panneer

Member

Registered: 2006-06-01

Posts: 110

Re: Please Please....

Thanks a lot...

Is anyother function except peicewise function?

---

Letter, number, arts and science
of living kinds, both are the eyes.

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: Please Please....

I don't get it. What does it mean to satisfy the conclusion but not the hypothosis?

---

A logarithm is just a misspelled algorithm.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Please Please....

If A then B

A is the premise, B is the conclusion.  However, it is possible for A to be false and B to be true.

If x is 3, then x is odd.

"x is 3" is the premise, "x is odd" is the conclusion.  x=5 satisfies the conclusion, but not the premise.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: Please Please....

mmm... i see..

well I'd say a function thats naturally discontinuous is more elegant. Such as y = 1/x from - 1 to 2.

---

A logarithm is just a misspelled algorithm.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Please Please....

Any non-smooth function won't satisfy the premise.  After this, it's just a matter of finding the right interval.  It just happens that most functions we know of are smooth.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Prakash Panneer

Member

Registered: 2006-06-01

Posts: 110

Re: Please Please....

Non smooth function???

Can you give me an example?

---

Letter, number, arts and science
of living kinds, both are the eyes.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Please Please....

Any function which isn't continuous on [a, b] or differentiable on (a, b) is not smooth.

1/x on [-1, 1] works.  So does just about any piecewise function (although not all).  |x| from [-1, 1] works as well.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."