Practicing Success
Let $f$ be a real-valued function defined on the interval $(0, \infty)$ by $f(x)=\ln x+\int\limits_0^x \sqrt{1+\sin t} d t$. Then which of the following statement(s) is(are) true? (a) $f^{\prime \prime}(x)$ exists for all $(x \in(0, \infty)$ |
(a), (b) (b), (c) (c), (d) (a), (d) |
(b), (c) |
We have, $f(x) =\ln x+\int\limits_0^x \sqrt{1+\sin t} d t$ $\Rightarrow f^{\prime}(x) =\frac{1}{x}+\sqrt{1+\sin x}$ $\Rightarrow f^{\prime}(x)$ exists for all $x \in(0, \infty)$ and is continuous on $(0, \infty)$ Clearly, $f^{\prime}(x)$ is not differentiable at $x=2 n \pi-\frac{\pi}{2}, n \in N$. So, $f^{\prime}(x)$ is not differentiable on $(0, \infty)$. |