Obtain the set of real numbers c Show that there exists a positive real number x!=2 such that log2x=x/2 . Hence obtain the set of real numbers c such that (log_2x)/x=c has only one real solution.

PoentWeptgj

PoentWeptgj

Answered question

2022-07-18

Obtain the set of real numbers c
Show that there exists a positive real number x 2 such that log 2 x = x 2 . Hence obtain the set of real numbers c such that log 2 x x = c has only one real solution.

Answer & Explanation

Jaylene Tyler

Jaylene Tyler

Beginner2022-07-19Added 10 answers

Hint
If you consider the function
f ( x ) = log 2 x x c
and compute its derivative, you find
f ( x ) = 1 log ( x ) x 2 log ( 2 )
which cancels for x = e. The second derivative test shows that this corresponds to a maximum. Now, you have
f ( e ) = 1 e log ( 2 ) c
So, for 1 < x < there are two solutions is c < 1 e log ( 2 ) , the two roots are identical if c = 1 e log ( 2 ) and there is no root for c > 1 e log ( 2 ) . So a single solution happens for 0 < x < 1
I am sure that you can take from here.
Ethen Frey

Ethen Frey

Beginner2022-07-20Added 6 answers

Hint: the function
x log 2 x x
is increasing from 0 to e and decreasing from e to + (why?). Also,
lim x 0 + log 2 x x =
lim x + log 2 x x =

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