Prove that this function is nonnegative

  • Thread starter sergey_le
  • Start date
  • #1
69
14

Homework Statement:

Prove that ##x^2##/2>xcosx-sinx for all x≠0

Relevant Equations:

-
What I wanted to do was set f(x)=##x^2##/2 - xcosx+sinx And show that f(x)>0.
f'(x)=x(1+sinx)
First I wanted to prove that f(x)<0 in the interval (0,∞)
0≤1+sinx≤2
And thus for all x> 0 f'(x)≥0 and therefore f(x)≥f(0)=0
And it doesn't help me much because I need to f(x)>0
 

Answers and Replies

  • #2
34,156
5,777
Homework Statement:: Prove that ##x^2##/2>xcosx-sinx for all x≠0
Homework Equations:: -

What I wanted to do was set f(x)=##x^2##/2 - xcosx+sinx And show that f(x)>0.
f'(x)=x(1+sinx)
First I wanted to prove that f(x)<0 in the interval (0,∞)
No, you need to prove that f(x) > 0 in that interval, just as you say above.
sergey_le said:
0≤1+sinx≤2
And thus for all x> 0 f'(x)≥0 and therefore f(x)≥f(0)=0
And it doesn't help me much because I need to f(x)>0
You showed that ##f'(x) \ge 0## for ##x \in [0, \infty)##, which means that the graph of f is increasing on this interval. For the other part of this problem, show that ##f'(x) \le 0## for ##x \in (-\infty, 0]##. Do you see how this helps you with the other part of the problem?
 
  • Like
Likes FactChecker, sergey_le and Math_QED
  • #3
69
14
No, you need to prove that f(x) > 0 in that interval, just as you say above.
You showed that ##f'(x) \ge 0## for ##x \in [0, \infty)##, which means that the graph of f is increasing on this interval. For the other part of this problem, show that ##f'(x) \le 0## for ##x \in (-\infty, 0]##. Do you see how this helps you with the other part of the problem?
No
That's what I intended to do.
But my problem is with the one that can be x∈(0,δ) (when are δ>0) so that f'(x)=0 and then f(0)=f(x)=0 for ∀x∈(0,δ) And then what I have to prove is wrong.
Do you understand my problem?
 
  • #4
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
14,720
6,968
No
That's what I intended to do.
But my problem is with the one that can be x∈(0,δ) (when are δ>0) so that f'(x)=0 and then f(0)=f(x)=0 for ∀x∈(0,δ) And then what I have to prove is wrong.
Do you understand my problem?
Try to prove:

If ##f(0)=0## and ##f'(x) > 0## for all ##x > 0## then ##f(x) > 0## for all ##x >0##. Even if ##f'(0)=0##.
 
  • Like
Likes sergey_le
  • #5
FactChecker
Science Advisor
Gold Member
5,787
2,155
Do you know the Fundamental Theorem of Calculus? That is a good way to use properties of the derivative to prove things about the function.
 
  • #6
69
14
Try to prove:

If ##f(0)=0## and ##f'(x) > 0## for all ##x > 0## then ##f(x) > 0## for all ##x >0##. Even if ##f'(0)=0##.
If that was the case then I had no problem .
But because it is ≤ Then there is the possibility that f'(x)=0.
 
  • #7
69
14
Do you know the Fundamental Theorem of Calculus? That is a good way to use properties of the derivative to prove things about the function.
Yes but I can't use it in this course.
In my course it is Calculus 2
 
  • #8
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
14,720
6,968
If that was the case then I had no problem .
But because it is ≤ Then there is the possibility that f'(x)=0.
Why did you choose ##\le##?

What I mean is this: suppose you wanted to show that ##z > 0##, say. First you show that ##z \ge 0## and then say you are stuck. So, you need to go back and try to prove that ##z > 0##.
 
Last edited:
  • Like
Likes sergey_le
  • #9
FactChecker
Science Advisor
Gold Member
5,787
2,155
Yes but I can't use it in this course.
In my course it is Calculus 2
Are you allowed to use the Mean Value Theorem?
 
  • Like
Likes sergey_le
  • #10
69
14
Why did you choose ##\le##?

What I mean is this: suppose you wanted to show that ##z > 0##, say. First you show that ##z \ge 0## and then say you are stuck. So, you need to go back and try to prove that ##z > 0##.
That's exactly my problem. Don't see a reason that z> 0
 
  • #12
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
14,720
6,968
That's exactly my problem. Don't see a reason that z> 0
Okay, let's start at the beginning. You have defined the function:

##f(x) = \frac 1 2 x^2 -x\cos x + \sin x##

You want to show that ##\forall x \ne 0: \ f(x) > 0## Let's start with ##x > 0##

Note that ##f(0) = 0##. First, you have to differentiate ##f(x)##. Can you do that?
 
  • Like
Likes sergey_le
  • #13
FactChecker
Science Advisor
Gold Member
5,787
2,155
I think that you can use the Mean Value Theorem to prove it. Remember that the slope of a tangent line is the derivative. Use a proof by contradiction. Assume that there is a point, a, where f(a)<0 and show that the assumption must be wrong.
 
  • #14
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
14,720
6,968
I think that you can use the Mean Value Theorem to prove it. Remember that the slope of a tangent line is the derivative. Use a proof by contradiction. Assume that there is a point, a, where f(a)<0 and show that the assumption must be wrong.
That almost works, except it must be shown that ##f(x) > 0##.
 
  • #15
69
14
I think that you can use the Mean Value Theorem to prove it. Remember that the slope of a tangent line is the derivative. Use a proof by contradiction. Assume that there is a point, a, where f(a)<0 and show that the assumption must be wrong.
If I use contradiction, So I have to assume that f(a)≤0 , And I don't see a reason why that f(a)=0
 
  • #16
69
14
Okay, let's start at the beginning. You have defined the function:

##f(x) = \frac 1 2 x^2 -x\cos x + \sin x##

You want to show that ##\forall x \ne 0: \ f(x) > 0## Let's start with ##x > 0##

Note that ##f(0) = 0##. First, you have to differentiate ##f(x)##. Can you do that?
I can show that f(x)≥0 .
 
  • #17
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
14,720
6,968
I can show that f(x)≥0 .
I know that. And you're asked to show that ##f(x) > 0##.

Why don't you differentiate the function? We need to analyse the derivative.
 
  • #18
69
14
I know that. And you're asked to show that ##f(x) > 0##.

Why don't you differentiate the function? We need to analyse the derivative.
Also the derivative f'(x)≥0
I don't understand what you want me to do?
How do I show that f'(x)>0 And no ≥
 
  • #19
34,156
5,777
First, you have to differentiate f(x)f(x)f(x). Can you do that?
Why don't you differentiate the function?
The OP did that in post #1, quoted below.

f'(x)=x(1+sinx)
 
  • #20
69
14
The OP did that in post #1, quoted below.

f'(x)=x(1+sinx)
I'm sorry I don't understand what you want to say
 
  • #21
34,156
5,777
I'm sorry I don't understand what you want to say
@PeroK asked you if you could differentiate the function. He must not have noticed that you showed this in post #1.
 
  • Like
Likes sergey_le
  • #22
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
14,720
6,968
Also the derivative f'(x)≥0
I don't understand what you want me to do?
How do I show that f'(x)>0 And no ≥
When is ##f'(x) = 0##?

What happens when ##f'(x) = 0##?

Analyse the derivative!
 
  • #23
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
14,720
6,968
The OP did that in post #1, quoted below.

f'(x)=x(1+sinx)
Yes, I did miss that. Nevertheless I did say in post #12 that we should start at the beginning!
 
  • #24
34,156
5,777
When is ##f'(x) = 0##?
What happens when ##f'(x) = 0##?
Yes, and @sergey_le, what are the values of f(x) at the points where f'(x) = 0?
 
  • Like
Likes sergey_le
  • #25
69
14
Yes, and @sergey_le, what are the values of f(x) at the points where f'(x) = 0?
nothing special.
Please direct me
 

Related Threads on Prove that this function is nonnegative

Replies
10
Views
3K
  • Last Post
Replies
3
Views
1K
Replies
7
Views
5K
Top