Question

Find the points of inflection and discuss the concavity of the graph of the function.

f (x) =  x^3 - 6x^2 + 12x

Answer 1

Step 1:

The function is

Differentiate the function with respect to .

Differentiate with respect to .

Find the inflection point by equating

Answer 2

Step 2:

Thus the inflection point is , Split the interval into , and .

Interval	Test Value	Sign of	Concavity
			Down
			Up

Thus the graph is concave down in the interval and concave up in the interval .

Inflection points:

Inflection point .

Solution:

Inflection point .

The graph is concave down in the interval

The graph is concave up in the interval .