226. Some inequalities from Calculus (Real Analysis).
by Virgil Nicula, Feb 19, 2011, 11:03 AM
PP 1. Show that 
for any 
.
Method 1.
. Denote 
. Therefore,
is strictly
increasing also
is strictly increasing also 
is
strictly increasing also
is strictly increasing also 
, i.e
. See here another problems.
Method 2.![$f(x)=\frac{\sin x}{\sqrt[3]{\cos x}}-x\ ,\ x\in\left(0,\frac {\pi}{2}\right)\implies$](//latex.artofproblemsolving.com/c/3/f/c3f928ba1812435934e4cfedd0dc2e0fb328b400.png)
![$f'(x)=\frac{1+2\cos^2x-3\sqrt[3]{\cos^4x}}{3\sqrt[3]{\cos^4x}}>0$](//latex.artofproblemsolving.com/e/a/6/ea60338ea35ee0b94f6cc4060751e55984953e18.png)
by 
..
Application. Prove that
for any 
and 
(Murray Klamkin, Crux Mathematicorum).
Remark. This inequality is a particular case of the inequality from here.
Proof. From
obtain that 
.
Remain to show that
, i.e. 
for any 
, what is here.
PP 2. Ascertain
, where ![$I=\left(0,\frac {\pi}{2}\right]$](//latex.artofproblemsolving.com/a/8/a/a8abc00f9ba0fb210e7d7ee466112d4d2d4d4871.png)
and the correspondence is 
.
Proof. I"ll study the monotony of
. Thus 
![$\left[x-(1-\cos x)\cdot \frac{1+\sqrt{5+4\cos x}}{2\sin x}\right] \equiv\ g(x)$](//latex.artofproblemsolving.com/2/6/9/2699852ba5c6dd63a42f0c7645937338eb3e48c5.png)
.
Observe that
and 
. Thus, 
and 
. In conclusion, 
for any ![$x\in \left(\ 0\ ,\ \frac {\pi}{2}\ \right]$](//latex.artofproblemsolving.com/1/6/3/16300d56f11f51a696e44576b00e85be5f4177ae.png)
.
PP3. Prove that for any positive numbers
, 
and for any 
exists the inequality 
.
Proof. Suppose w.l.o.g. that
. Thus, need to prove that for any 
and for any exists the relation 
. Indeed, 
. Let 
. Hence, it remains to prove that 
for 
, where 
, which is true because 
.
See here
for any 
.Method 1.
. Denote 
. Therefore,
is strictlyincreasing also
is strictly increasing also 
isstrictly increasing also
is strictly increasing also 
, i.e
. See here another problems.Method 2.
![$f(x)=\frac{\sin x}{\sqrt[3]{\cos x}}-x\ ,\ x\in\left(0,\frac {\pi}{2}\right)\implies$](http://latex.artofproblemsolving.com/c/3/f/c3f928ba1812435934e4cfedd0dc2e0fb328b400.png)
![$f'(x)=\frac{1+2\cos^2x-3\sqrt[3]{\cos^4x}}{3\sqrt[3]{\cos^4x}}>0$](http://latex.artofproblemsolving.com/e/a/6/ea60338ea35ee0b94f6cc4060751e55984953e18.png)
by 
..Application. Prove that
for any 
and 
(Murray Klamkin, Crux Mathematicorum).Remark. This inequality is a particular case of the inequality from here.
Proof. From
obtain that 
.Remain to show that
, i.e. 
for any 
, what is here.PP 2. Ascertain
, where ![$I=\left(0,\frac {\pi}{2}\right]$](http://latex.artofproblemsolving.com/a/8/a/a8abc00f9ba0fb210e7d7ee466112d4d2d4d4871.png)
and the correspondence is 
.Proof. I"ll study the monotony of
. Thus 
![$\left[x-(1-\cos x)\cdot \frac{1+\sqrt{5+4\cos x}}{2\sin x}\right] \equiv\ g(x)$](http://latex.artofproblemsolving.com/2/6/9/2699852ba5c6dd63a42f0c7645937338eb3e48c5.png)
.Observe that
and 
. Thus, 
and 
. In conclusion, 
for any ![$x\in \left(\ 0\ ,\ \frac {\pi}{2}\ \right]$](http://latex.artofproblemsolving.com/1/6/3/16300d56f11f51a696e44576b00e85be5f4177ae.png)
.PP3. Prove that for any positive numbers
, 
and for any 
exists the inequality 
.Proof. Suppose w.l.o.g. that
. Thus, need to prove that for any 
and for any exists the relation 
. Indeed, 
. Let 
. Hence, it remains to prove that 
for 
, where 
, which is true because 
.See here
This post has been edited 31 times. Last edited by Virgil Nicula, Jan 7, 2018, 5:02 PM
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