### Home > CALC > Chapter 7 > Lesson 7.1.5 > Problem 7-53

Use the first and second derivatives to determine the following locations for

*f*(*x*) =*xe*. Homework Help ✎^{x}Relative minima and maxima.

Intervals at which

*f*(*x*) is increasing and decreasing.Inflection points.

Intervals at which

*f*(*x*) is concave up and concave down.

Remember: Finding where *f* '(*x*) = 0 or *f* '(*x*) = DNE will identify CANDIDATES for minima and maxima.

You need to complete the 1st or 2nd Derivative Test to confirm which is which.

When *f* '(*x*) is positive, then *f*(*x*) has positive slopes, which means *f*(*x*) is increasing.

Remember: Finding where *f* ''(*x*) = 0 or *f* ''(*x*) = DNE will identify CANDIDATES for inflection points.

You need to do further investigation to determine if it is (or is not) an inflection point.

Either test for a sign change of *f* ''(*x*) before and after the candidate point. Or evaluate *f* '''(*x*) at the candidate.

If *f* '''(candidate) ≠ 0, then it is a point of inflection.

When *f* ''(*x*) is positive, then *f* '(*x*) has positive slopes, which means *f*(*x*) is concave up.