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.).

26. \(f\left( x \right) = {e^x}\)

Short answer

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

There is no local as well as an absolute maximum.

There is no local as well as an 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:

\(f\left( x \right) = {e^x}\)

The graph is plotted as:

The absolute and local maximum or minimum values cannot be determined for any exponential graph.

Hence, there is no local as well as an absolute maximum. Also, there is no local as well as an absolute minimum.