If a function R → R is strictly increasing and differentiable and it is true that lim x → ∞ f ( x ) = l where l ∈ R , then f is concave at some interval of the form ( k , + ∞ ).

Question

If a function       R    →      R   is strictly increasing and differentiable and it is true that       lim          x      →      ∞        f  (  x  )  =  l where   l  ∈      R  , then   f is concave at some interval of the form   (  k  ,  +  ∞  ).

lelestalis80d · Accepted Answer

Counterexample: Define  f  (  x  )  =      ∫    0    x              2      +      sin      ⁡      t              1      +              t        2                d  t  .Then   f  ∈      C    ∞    (      R    )  .. Note that the above integral converges if   x  =  ∞  ,, showing that       lim          x      →      ∞        f  (  x  ) exists. By the FTC,       f    ′    (  x  )  =  (  2  +  sin  ⁡  x  )      /    (  1  +      x    2    )  &amp;gt;  0  ,, so f is strictly increasing. Differentiating again, we get                    (1)                              f          ″                (        x        )        =                              (            1            +                          x              2                        )            cos            ⁡            x            −            (            2            +            sin            ⁡            x            )            (            2            x            )                                (            1            +                          x              2                                      )              2                                      .            Along the sequence   2  n  π  ,, the numerator in   (  1  )  →  ∞  ,, so       f    ″    (  2  n  π  )  &amp;gt;  0 for large   n  ..

Paula Cameron · Accepted Answer

t is not true. Think for example at :  x  ↦  exp  ⁡  (  −  x  )  sin  ⁡  (  x  )

If a function f : R->R is strictly increasing and differentiable and it is true that lim_(x->oo) f(x)=l where l in R, then f is concave at some interval of the form (k,+oo).

Answered question

Answer & Explanation

New Questions in Calculus 1