- #1
anemone
Gold Member
MHB
POTW Director
- 3,883
- 115
Hi MHB,
When I first saw the problem (Prove that $\sin^2 x<\sin x^2$ for $0\le x\le \sqrt{\dfrac{\pi}{2}}$), I could tell that is one very good problem, but, a good problem usually indicates it is also a very difficult problem and after a few trials using calculus + trigonometry method, I failed and the only conclusion that I could draw by now is one cannot use calculus route in this problem. (But I could be wrong.)
So, I decided to post it here and I would appreciate it if you could tell me what method you would use in your attempt, thanks so much in advance.
Edit:
Oops..now that I redo the problem and I noticed I could use some trick in the calculus method that I overlooked before...so, I finally proved it! Yeah! I will post my solution later.
But, I still hope to see how others would attempt at the problem and I would deeply appreciate it if you could share your solution with me too!
When I first saw the problem (Prove that $\sin^2 x<\sin x^2$ for $0\le x\le \sqrt{\dfrac{\pi}{2}}$), I could tell that is one very good problem, but, a good problem usually indicates it is also a very difficult problem and after a few trials using calculus + trigonometry method, I failed and the only conclusion that I could draw by now is one cannot use calculus route in this problem. (But I could be wrong.)
So, I decided to post it here and I would appreciate it if you could tell me what method you would use in your attempt, thanks so much in advance.
Edit:
Oops..now that I redo the problem and I noticed I could use some trick in the calculus method that I overlooked before...so, I finally proved it! Yeah! I will post my solution later.
But, I still hope to see how others would attempt at the problem and I would deeply appreciate it if you could share your solution with me too!
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