Let the function f be defined by f ( x ) = x ln ⁡ x, for all x &gt; 0. Then A)f is increasing on ( 0 , e − 1 ); B)f is decreasing on (0,1); C)The graph of f is concave down for all x; D)The graph of f is concave up for all x

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Let the function f be defined by   f  (  x  )  =  x  ln  ⁡  x, for all x &amp;gt; 0. Then A)f is increasing on (  0  ,      e          −      1      ); B)f is decreasing on (0,1); C)The graph of f is concave down for all x;  D)The graph of f is concave up for all x

Scarlet Barrett · Accepted Answer

The correct answer is  C All x cause the f graph to be concave up.            d      y              d      x        =  1  +  ln  ⁡  x              d      y              d      x        =  0  ⇒  x  =      e          −      1        =      1    e    ⇒ It is increasing on   (      1    e    ,  ∞  ) and decreasing on   (  0  ,      1    e    ) Clearly,   x  =      1    e   gives local minima  and   f  (      1    e    )  =  −      1    e      Also,                     d        2            y              d              x        2              =      1    x    &amp;gt;  0  (  ∵  x  &amp;gt;  0  )  ⇒ Concave up for all x &amp;gt; 0

Let the function f be defined by f(x)=x ln x, for all x > 0. Then A)f is increasing on (0,e^(−1)); B)f is decreasing on (0,1); C)The graph of f is concave down for all x; D)The graph of f is concave up for all x

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