Let h be a twice differentiable positive function on an open interval J. Let g(x)=ln(h(x)) for each x∈J. Suppose ( h ′ ( x ) ) 2 &gt; h ′ ′ ( x ) h ( x ) for each x∈J. Then A) g is increasing on JB) g is decreasing on JC) g is concave up on J D) g is concave down on J

Question

Let h be a twice differentiable positive function on an open interval J. Let g(x)=ln(h(x)) for each x∈J. Suppose   (  h      ′    (  x  )      )    2    &amp;gt;  h      ′        ′    (  x  )  h  (  x  ) for each x∈J. Then A) g is increasing on JB) g is decreasing on JC) g is concave up on J D) g is concave down on J

Jewel Kelly · Accepted Answer

The correct answer is  D: g is concave down on Jg(x)=ln(h(x))  g      ′    (  x  )  =            h              ′            (      x      )              h      (      x      )      Again differentiating w.r.t. x,  g  &quot;  (  x  )  =            h      (      x      )      h      &quot;      (      x      )      −      (              h        ′            (      x      )              )        2                            h        2            (      x      )        (given)  ∴  g  &quot;  (  x  )  &amp;lt;  0⇒g(x) is concave down.g(x) depending on the sign of can be increasing or decreasing h′(x).

Let h be a twice differentiable positive function on an open interval J.

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