Question

The equation $x\log x=3-x$ has at least one root in the interval

Answer 1

Step 1: Assume $y=x\log x$ and $y=3-x$ and then draw the graph of both functions

We will have to draw the graphs of functions $y=x\log x$ and $y=3-x.$

The solution of the equation: $x\log x=3-x$will be given by the point of intersection of the two graphs.

Step 2: Identify the point where both the graph intersects

Consider the graph of $y=x\log x$

The domain of this function is $(0,\infty )$since $\log x$ is defined for $x>0.$

$$\begin{gathered} lety=0 \\ \Rightarrow x\log x=0 \\ \Rightarrow x=0orx=1 \end{gathered}$$

For, $y=3-x,$ the graph is a straight line with $x-$intercept $3$ and $y-$intercept $3$.

From the graph shown above, we can see that the two functions intersect at a point between

$x=1andx=3.$

The equation $x\log x=3-x$ has at least one root in the interval in $(1,3).$

Hence, the required interval is $(1,3).$