Finding the maximum value without using derivativesFind 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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Finding the maximum value without using derivativesFind 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.

Jazmin Bryan · Accepted Answer

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.

Andreasihf · Accepted Answer

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.

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.

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