Practicing Success
$f(x)=\int\limits_1^x \frac{e^t}{t} d t, x \in R^{+}$. Then complete set of values of x for which f(x) ≤ ln x is |
(0, 1] [1, ∞) (0, ∞) None of these |
(0, 1] |
$f(x)=\int\limits_0^x \frac{e^t}{t} d t \Rightarrow f(1)=0$ and $f'(x)=\frac{e^x}{x}$ Let $g(x)=f(x)-\ln (x) . x \in R^{+}$ $\Rightarrow g'(x)=f'(x)-\frac{1}{x}=\frac{e^x-1}{x}>0 ~\forall~ x \in R^{+}$ ⇒ g(x) is increasing for $\forall~ x \in R^{+}$ $g(1)=f(1)-\ln 1=0-0=0$ $\Rightarrow g(x)>0 ~\forall~ x>1$ and $g(x) \geq 0 ~\forall~ x \in(0,1]$ $\Rightarrow \ln x \geq f(x) \forall x \in(0,1]$ |