Question

Find the extrema and the points of inflection (if any exist) of the function. use a graphing utility to graph the function and confirm your result.

Answer 1

Step 1:

The function  .

Consider  .

Differentiate with respect to .

Derivative of the exponential function .

.

Step 2:

Find extrema by equating the first derivative to zero.

Substitute the value in original function.

The function has extrema at .

Step 3:

Determine nature of the extrema, using second derivative test.

Apply derivative with respect to .

Point	Sign of

The absolute maximum at .

Answer 2

Continued:

Step 4:

For inflection points, equate second derivative to zero.

Inflection points:

Inflection points are .

Step 5:

Graph:

Observe the graph the function has the absolute maximum at .

Solution:

The function has absolute maximum at .