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5-43.
  1. Using the graph of the slope function at right, determine where the following situations occur for g : Homework Help ✎

    1. Relative minima.

    2. Relative maxima.

    3. Interval(s) over which g is increasing.

    4. Inflection points.

    5. Interval(s) over which g is concave up.

    6. Interval(s) over which g is concave down.

Notice that this is a graph of g′(x), but you are being asked about y = g(x).

Relative minima and maxima can exist when g′(x) = 0.
This occurs when x = a, x = d, and x = f. But which points indicate a maximum and which indicate a minimum?

A minimum occurs when the slope changes from negative to positive.

See the hint in part (a). A maximum occurs when the slope changes from positive to negative.

g is increasing when g′(x) > 0.

Inflection points can happen where g″(x) = 0. Remember, we are looking at the graph of y = g′(x).

g is concave up when g″(x) > 0.

g is concave down when g′(x) is decreasing.