Question

Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f\). (Use the graphs and transformations of Sections 1.2 and 1.3.).

\(f\left( x \right) = \ln x,\,\,\,\,\,\,0 < x \le 2\)

Short answer

The graph of \(f\left( x \right)\)is:

There is no local maximum.

The absolute maximum is \(f\left( 2 \right) = 0.69\).

There is no local as well as the absolute minimum.

Step-by-step solution

Minima and Maxima of a function

TheCritical numbers for any function \(f\left( x \right)\) can be obtained by putting \(f'\left( x \right) = 0\).

For the points \(a{\rm{ and }}b\)such that:

\(\begin{array}{l}f\left( a \right) \to {\rm{maximum}} \Rightarrow a\,\,{\rm{is maxima}}\\f\left( b \right) \to {\rm{minimum}}\,\,\, \Rightarrow a\,\,{\rm{is minima}}\end{array}\)

Graphing the function for minima and maxima:

The given function is:

\(f\left( x \right) = \ln x,\,\,\,\,\,\,0 < x \le 2\)

Here, \(f\left( x \right)\) is continuous in the interval \(\left( {0,\left. 2 \right)} \right.\). So, the graph can be plotted as:

The absolute maximum value can be seen in the graph within the domain\(\left( {0,\left. 2 \right)} \right.\). There are no local as well as minimum values. Also, there is no local maximum here.

At \(x = 2\),

\(\begin{array}{c}f\left( x \right) = \ln 2\\ = 0.69\end{array}\)

This is the absolute maximum value.

Hence, there is no local maximum. The absolute maximum is \(f\left( 2 \right) = 0.69\).

There is no local as well as an absolute minimum for this function.