f(x)=(x+2)^3 -4 The point (-2,-4) is which of the following?

Question

How do you solve this problem? Which is the right answer and why?

a. An absolute maximum

b. An absolute minimum

c. A critical point but not an extremum

d. Not a critical point

e. none of these

Answer 1

Differentiation gives f'(x) = 3(x+2)² which is 0 at x = -2. Therefore this is a critical point. Notice that the derivative is positive everywhere else. Therefore this point is not an extremum. So the answer is c.

Answer 2

I agree with Roman using and his differentiation, however, this is not a Calculus question. From Algebra 1, f(x) is in the form of y = x^3 which passes through the point (0,0) and changes the instantaneous slope from positive to negative, it is a point of inflection. 

As expressed f(x) = (x+2)^3 - 4 tells us the translation of the y = x^3 graph with its inflection point of (0,0) to a new inflection point of (-2,-4). c. a critical point but not an extremium