Let `f(x) = sqrt(log_(10)x^(2))`.Find the set of all values of x for which f (x) is real.

Options

A. `(0,1]`
B. `[1,oo)`
C. `(0,oo)`
D. none of these

Answer

A

Explanation

`f(x)=int_(1)^(x) (e^(t))/tdtimpliesf(1)=0` and `f'(x)=(e^(x))/x`
Let `g(x)=f(x)-In (x),x epsilonR^(+)`
`:.g'(x)=f'(x)-1/x=(e^(x)-1)/x gt 0 AA x epsilonR^(+)`
Thus `g(x)` is increasing for `x epsilonR^(+)`
`g(1)=f(1)-In1=0-0=0`
`:. g(x)gtAAxgt1` and `g(x)le0AAxepsilon(0,1]`
`:.In x ge f(x)AA x epsilon (0,1]`