What is the value of c for which the instantaneous rate of change of f at $x=c$ is the same as the average rate of change f over [1,4] if $f(x)=x+lnx$ ?

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Answer 1 · 2015-03-27T16:00:15

$c=3/ln4$

$f(x)=x+lnx$

Average rate of change over ${1, 4}$ is:

$(Delta f)/(Delta x)=(f(4)-f(1))/(4-1)=(4+ln4-(1+ln1))/3=(3+ln4)/3$

$f'(x)=1+1/x$

$c$ is the solution to
$1+1/x=(3+ln4)/3$

$1/x = ln4 / 3$

$x= 3/ ln4$

What is the value of c for which the instantaneous rate of change of f at x=cx=c is the same as the average rate of change f over [1,4] if f(x)=x+lnxf(x)=x+lnx?