Question

Consider the function y=[(x-7)x]^2

1. At what value(s) of x is it possible that we have a local maximum or minimum, or an inflection point?

2. Use the general test to determine whether we have a maximum, minimum, or inflection point, for each of the critical values you found in part (1).

Answer 1

The function is given as or .

1. The FOC is or or or , ie or .

Hence, the required critical values are 0, 3.5 and 7, where we would have either maximum or minimum.

2. We have or or .

At the critical points, we have or , or or , and or .

Hence, the function has minimum at x=0,7, and a maximum at x=3.5.

Note : There are non-stationary inflection points for the x's where , even though at those points. Those aren't included since the evaluation of extremum and inflection points are asked only for critical points.