418. Some nice problems with polynomials/equations III
by Virgil Nicula, Mar 28, 2015, 2:33 PM
PP6. Solve the equation ![$\sqrt[4]{1-x^4}=\sqrt[5]{1-x^5}$](//latex.artofproblemsolving.com/1/3/7/1379078647011f6b231204195433f7a45868f725.png)
.
Proof 1. Prove easily that![$x\in [0,1]$](//latex.artofproblemsolving.com/5/6/1/561b0f8577e618149adc3fae918f20a4a4d4c5ca.png)
and 
- the solution of the given equation. Suppose w.l.o.g. 
. Therefore, ![$\sqrt[4]{1-x^4}=\sqrt[5]{1-x^5}\iff$](//latex.artofproblemsolving.com/f/d/2/fd2113a01eaa25d3d7aed0b4c961d5a5cc91644d.png)
, where 
. So 
is (strict) increasing and 
for any . Hence 
.
Proof 2. Since and - solution of given equation, can suppose w.l.o.g. . Thus, ![$\sqrt[5]{1-x^5}>\sqrt[4]{1-x^4}\iff$](//latex.artofproblemsolving.com/b/d/7/bd71ad24532ec4d9a4a2f3cfa903a2371fa31f49.png)
, where 
, what is truly because 
for any .
PP7. Solve the irrational equation
, where 
.
Proof. Let
. Obtain that 
. Thus, 
. Obtain analogously get the relations 
and 
. Therefore, 
and
and 
, i.e. 
.
PP8. Solve the equation
.
Proof 1.
.
Proof 2. Observe that
and 
and I"ll use the substitution
. Thus, ![$27\left[\left(x+\frac 1x\right)-2\right]=4\left[\left(x+\frac 1x\right)-1\right]^3\iff$](//latex.artofproblemsolving.com/8/5/b/85b6435483ceec431033bb0ff064f0e4d0dba899.png)
.
PP9. Prove that for any
we have 
.
Remark.![$\boxed{\ x\in [a,b]\cup [b,a]\iff (x-a)(x-b)\le 0\iff \left|x-\frac {a+b}{2}\right|\le\left|\frac {a-b}{2}\right|\iff |x-a|+|x-b|=|a-b|\ }$](//latex.artofproblemsolving.com/e/e/5/ee563af3aa280eb8490ce356d37a0cb2b9d26f36.png)
Proof 1.
, abs.
Proof 2. 
![$\left[\left(3x^2-10x+11\right)-\left(x^2+2x-7\right)\right]\cdot $](//latex.artofproblemsolving.com/c/e/c/cec998d0d3dcaaab15b8a94e013050fc6fee2bca.png)
![$\left[\left(3x^2-10x+11\right)+\left(x^2+2x-7\right)\right]<0\iff$](//latex.artofproblemsolving.com/b/a/7/ba7d7334e816e21ec99583ac87cc4ecc254364fe.png)
, abs.
PP10. Find the tangent line which touches the curve
at distinct two points.
Proof 1. The line
is tangent to the curve 
iff the equation 
has two
solutions, each with multiplicity 2, i.e.
. Obtain that 
, where 
and 
.
Proof 2. is tangent to 
and 
have at least two common roots, where 
. Thus,
=====================================================================
=====================================================================
=========================================================================
=========================================================================
========================================================
Thus,
.
In the second case the tangent points coicide with the point
, i.e. 
. Therefore, the answer of our problem is .
See here.
![$\sqrt[4]{1-x^4}=\sqrt[5]{1-x^5}$](http://latex.artofproblemsolving.com/1/3/7/1379078647011f6b231204195433f7a45868f725.png)
.Proof 1. Prove easily that
![$x\in [0,1]$](http://latex.artofproblemsolving.com/5/6/1/561b0f8577e618149adc3fae918f20a4a4d4c5ca.png)
and 
- the solution of the given equation. Suppose w.l.o.g. 
. Therefore, ![$\sqrt[4]{1-x^4}=\sqrt[5]{1-x^5}\iff$](http://latex.artofproblemsolving.com/f/d/2/fd2113a01eaa25d3d7aed0b4c961d5a5cc91644d.png)
, where 
. So 
is (strict) increasing and 
for any . Hence 
.Proof 2. Since
and - solution of given equation, can suppose w.l.o.g. . Thus, ![$\sqrt[5]{1-x^5}>\sqrt[4]{1-x^4}\iff$](http://latex.artofproblemsolving.com/b/d/7/bd71ad24532ec4d9a4a2f3cfa903a2371fa31f49.png)
, where 
, what is truly because 
for any .PP7. Solve the irrational equation
, where 
. Proof. Let
. Obtain that 
. Thus, 
. Obtain analogously get the relations 
and 
. Therefore, 
and
and 
, i.e. 
.PP8. Solve the equation
.Proof 1.
.Proof 2. Observe that
and 
and I"ll use the substitution
. Thus, ![$27\left[\left(x+\frac 1x\right)-2\right]=4\left[\left(x+\frac 1x\right)-1\right]^3\iff$](http://latex.artofproblemsolving.com/8/5/b/85b6435483ceec431033bb0ff064f0e4d0dba899.png)
.PP9. Prove that for any
we have 
.Remark.
![$\boxed{\ x\in [a,b]\cup [b,a]\iff (x-a)(x-b)\le 0\iff \left|x-\frac {a+b}{2}\right|\le\left|\frac {a-b}{2}\right|\iff |x-a|+|x-b|=|a-b|\ }$](http://latex.artofproblemsolving.com/e/e/5/ee563af3aa280eb8490ce356d37a0cb2b9d26f36.png)
Proof 1.
, abs.Proof 2.
![$\left[\left(3x^2-10x+11\right)-\left(x^2+2x-7\right)\right]\cdot $](http://latex.artofproblemsolving.com/c/e/c/cec998d0d3dcaaab15b8a94e013050fc6fee2bca.png)
![$\left[\left(3x^2-10x+11\right)+\left(x^2+2x-7\right)\right]<0\iff$](http://latex.artofproblemsolving.com/b/a/7/ba7d7334e816e21ec99583ac87cc4ecc254364fe.png)
, abs.PP10. Find the tangent line which touches the curve
at distinct two points.Proof 1. The line
is tangent to the curve 
iff the equation 
has twosolutions, each with multiplicity 2, i.e.
. Obtain that 
, where 
and 
.Proof 2.
is tangent to 
and 
have at least two common roots, where 
. Thus,=====================================================================
=====================================================================
=========================================================================
=========================================================================
========================================================
Thus,
.In the second case the tangent points coicide with the point
, i.e. 
. Therefore, the answer of our problem is .See here.
This post has been edited 14 times. Last edited by Virgil Nicula, Nov 12, 2015, 4:17 PM
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