Question

Find all roots, local maxima and minima, and inflection points of each function f . In addition, determine whether any local extrema are also global extrema on the domain of f .

$f\left(x\right)=x\ln \left(x\right)$.

Short answer

The roots of the function are $x=1$.

Local minima at $x=\frac{1}{e}$and it is also the global minima.

There are no local maximas.

There are no inflection points.

Step-by-step solution

Step 1. Given information

Given function is $f\left(x\right)=x\ln \left(x\right)$.

We have to find the roots, local maxima, local minima and inflection points of the function.

Step 2. Finding roots

The roots of a function are the values of $x$for which $f\left(x\right)=0$.

Here roots are:

$$\begin{gathered} x\ln \left(x\right)=0 \\ \Rightarrow x=0or\ln \left(x\right)=0 \\ \Rightarrow x=0,1 \end{gathered}$$.

Step 3. Finding Local maxima and local minima

Local maxima/ Local minima are the values of $x$for which $f\text{'}\left(x\right)=0$.

First derivate of the given function is:

$$\begin{gathered} f\left(x\right)=x\ln \left(x\right) \\ \Rightarrow \frac{df\left(x\right)}{dx}=\ln \left(x\right)\frac{dx}{dx}+x\frac{d\ln \left(x\right)}{dx} \\ \Rightarrow f\text{'}\left(x\right)=\ln \left(x\right)+1 \end{gathered}$$

Equating the first derivate to zero will give us:

$$\begin{gathered} 1+\ln \left(x\right)=0 \\ \Rightarrow \ln \left(x\right)=-1 \\ \Rightarrow x=\frac{1}{e} \end{gathered}$$

Step 4. Finding the second derivative

Let say $t$where $f\text{'}\left(t\right)=0$.

If $f\text{'}\text{'}(t>0)$then it is local minima.

If $f\text{'}\text{'}\left(t\right)<0$then it is local maxima.

The second derivative of the given function:

$$\begin{gathered} f\text{'}\left(x\right)=1+\ln \left(x\right) \\ \Rightarrow f\text{'}\text{'}\left(x\right)=\frac{1}{x} \end{gathered}$$

Substitute $x=\frac{1}{e}$in $f\text{'}\text{'}\left(x\right)=\frac{1}{x}$

$$\begin{gathered} f\text{'}\text{'}\left(\frac{1}{e}\right)=\frac{1}{{\displaystyle \frac{1}{e}}} \\ \Rightarrow f\text{'}\text{'}\left(\frac{1}{e}\right)=e \\ \Rightarrow f\text{'}\text{'}\left(\frac{1}{e}\right)>0 \end{gathered}$$

Therefore $x=\frac{1}{e}$is the local minima.

And it is also the global minima because the value of function increases for both$x>\frac{1}{e};x<\frac{1}{e}$.

Step 5. Inflection points

Inflection points are the points where $f\text{'}\text{'}\left(x\right)=0$.

For the given function:

$$\begin{gathered} f\text{'}\text{'}\left(x\right)=\frac{1}{x} \\ \Rightarrow \frac{1}{x}=0 \end{gathered}$$

Therefore there are no inflection points.