Question

Determine the open intervals on which the graph is concave upward or concave downward.

Answer 1

Step 1 :

Concavity test:

(a) If for all x  in I, then the graph of f  is concave upward on I.

(b) If for all x  in I, then the graph of f  is concave downward on I.

Step 2 :

The function is .

Differentiate  with respect to x.

Step 3 :

Differentiate  with respect to x.

Step 4 :

Determination of inflection points : 

Equate to zero.

Thus, the inflection points are .

Consider the test intervals as and .

Interval	Test Value	Sign of	Concavity
			Up
			Down
			Up

Thus, the graph is concave up on the interval and .

The graph is concave down on the interval .

Solution :

The function is concave up on the interval and .

The function is concave down 0n the interval .