`L e tf:[1,oo)vecRa n df(x)=int_1^x(e^t)/t dt-e^xdotT h e n` `f(x)` is an increasing function `("lim")_(xvecoo)f(x)vecoo` `f^(prime)(x)` has a maxima at `x=e` `f(x)` is a decreasing function

Options

A. `f(x)` is an increasing function
B. `lim_(x tooo) f(x)to oo`
C. `f'(x)` has a maxima at `x=e`
D. `f(x)` is a decreasing function

Answer

A, B

Explanation

`f(x)=x int_(1)^(x) (e^(t))/t dt-e^(x)`
`:.f'(x) = x(e^(x))/x+int_(1)^(x)(e^(t))/t dt-e^(x)`
`=int_(1)^(x)(e^(t))/t dt gt0 [ :' x epsilon[1,oo)]`
Thus `f(x)` is an increasing function.