Finding the maximum value without using derivatives Find the maximum value of f(x)=2 sqrt(x)- sqrt(x+1)- sqrt(x-1) without using derivatives.

Liam Keller 2022-09-13 Answered
Finding the maximum value without using derivatives
Find the maximum value of
f ( x ) = 2 x x + 1 x 1
without using derivatives.
The domain of f ( x ) is x [ 1 , ). Then, using derivatives, I can prove that the function decreases for all x from D ( f ) and the maximum value is f ( 1 ) = 2 2 . However, this uses derivatives.
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Answers (2)

Jazmin Bryan
Answered 2022-09-14 Author has 12 answers
Note that
f ( x ) = ( x x 1 ) ( x + 1 x ) = 1 x + x 1 1 x + 1 + x = x + 1 x 1 ( x + x 1 ) ( x + 1 + x ) = 2 ( x + x 1 ) ( x + 1 + x ) ( x + 1 + x 1 )
which is a decreasing function of x.

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Andreasihf
Answered 2022-09-15 Author has 3 answers
f ( x ) = 1 x + x 1 1 x + 1 + x =
= 2 ( x + x 1 ) ( x + x + 1 ) ( x + 1 + x 1 )
= 2 ( 1 + 2 ) 2 = 2 2 .
The equality occurs for x = 1, which says that we got a maximal value.

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