Problem: Determine concavity of the function

Solution: First we determine the domain. The cubic root accepts anything, so does a polynomial, so the domain is the whole real line. We start with one interval, now we will find out whether we have to split it further. The procedure calls for finding the second derivative.

We are looking for points of the domain at which the second derivative is zero or does not exist. It is zero at x = 0 and it does not exist for x = 1. Thus the domain splits into three intervals of concavity. To determine what kind we use a table. We will use closed endpoints when the function is continuous there. Note that the fifth power and the cubic root do not change the sign of the expression inside.

The given function is concave up on [0,1], concave down on (−∞,0] and on [1,∞). It has inflection points f (0) = 1 and f (1) = 0.

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