### Home > CALC > Chapter 5 > Lesson 5.1.4 > Problem 5-43

Using the graph of the slope function

*f ′*(*x*) below, determine where the following situations occur for*f*(*x*): Homework Help ✎

Relative minima and maxima.

Intervals at which

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

Intervals at which

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

Notice that this is a graph of *f* '(*x*), but you are being asked about *f*(*x*).

Relative minima and maxima can exist when *f* '(*x*) = 0. *x* = *a*, *x* = *d* and *x* = *f*. But which indicates a max and which indicates a min?

To distinguish between a relative minimum and a relative maximum, use the 1st or 2nd derivative test.

Refer to the hints in problem 5-42 for more guidance.

*f*(*x*) increases when the slope is positive... and we are looking at graph of the slopes.

Inflection points can happen where *f* ''(*x*) = 0... and we are looking at the graph of *f* '(*x*).

*f*(*x*) is concave up when *f* ''(*x*) > 0... and we are looking at a graph of *f* '(*x*).