Find the critical point of the following function.
$f (x) = x^{2 / 3} + 2 x^{- 1 / 3}$

Question

Find the critical point of the following function.  f  (  x  )  =      x          2              /            3        +  2      x          −      1              /            3

Simeon Hester · Accepted Answer

f          ′                (        x        )        =                  2          3                          x                      −            1                          /                        3                          −                  2          3                          x                      −            4                          /                        3                                              =                  2          3                          (                      x                          −              1                              /                            3                                −                      x                          −              4                              /                            3                                )                                                                =                  2          3                          (                      x                          3                              /                            3                                            x                          −              4                              /                            3                                −                      x                          −              4                              /                            3                                )                                                                =                  2          3                          x                      −            4                          /                        3                                    (                      x                          3                              /                            3                                −          1          )                                                                =                  2          3                          x                      −            4                          /                        3                          (        x        −        1        )                                                        =                              2            (            x            −            1            )                                3                          x                              4                                  /                                3                                                                                                    =                              2            (            x            −            1            )                                3                                          x                4                                            3                                                        Now, we have critical points where      f    ′    (  x  )  is undefined: at   x  =  0, as is   f  (  x  )      f    ′    (  x  )  =  0  ::  This will occur if and only when the numerator is zero. In this case,       f    ′    (  x  )  =  0 when   (  x  −  1  )  =  0    ⟹    x  =  1.

Nathanael Perkins · Accepted Answer

Hint:            x              −                  1          3                            x              −                  4          3                      =      x                  4        3            −              1        3              =  x    ∀  x  ≠  0.

Find the critical point of the following function. f(x)=x^(2/3)+2x^(−1/3)

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Answer & Explanation

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