Question

Consider the function $f\left(x\right)=\begin{array}{cc}\sqrt{x}\ln xwhen& x>0\\ 0for& x=0\end{array}$

Let the inflection points of the graph of $y=f\left(x\right)$ be $x=k$ .Find $k$?

Answer 1

$f\left(x\right)=\sqrt{x}\ln x$
$f^{\prime }\left(x\right)=\frac{1}{\sqrt{x}}+\frac{\ln x}{2\sqrt{x}}$
$f^{\prime \prime }\left(x\right)=-\frac{1}{2x\sqrt{x}}+\frac{1}{2}\frac{\frac{1}{\sqrt{x}}+\frac{\ln x}{2x\sqrt{x}}}{x}$
For point of inflection $f^{\prime \prime }\left(x\right)=0\Rightarrow x=1$

Since $x=k\to k=1$