For what values of a and b will the function defined by f(x)=a(x-b ln x)
have f(2)=5 as arelative minimum value.Justify your conclusion.
first 2 is a relative minimum for the function that mean it is a critical point for the first derivative
$\displaystyle f(x)=a(x-b\ln x) $
$\displaystyle f'(x) = a-\frac{ab}{x} $
$\displaystyle f'(2) = a-\frac{ab}{2} = 0 \Rightarrow b=2 $
and you have f(2)=5
$\displaystyle f(2) = 2a -ab\ln(2) =5$
$\displaystyle 2a-a(2)\ln(2) =5 \Rightarrow a=\frac{5}{2-2\ln (2)} $