f(x)=(x+2)^3-4. The point (-2, -4) is which of the following?
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
I think the answer is C, am I right?
Point of inflection
I agree with you
To determine if the point (-2, -4) is an absolute maximum, an absolute minimum, a critical point but not an extremum, or not a critical point for the function f(x) = (x+2)^3 - 4, we need to analyze the behavior of the function around the given point.
First, let's find the derivative of f(x) to determine the critical points:
f'(x) = 3(x+2)^2
To find the critical points, we set f'(x) = 0 and solve for x:
3(x+2)^2 = 0
(x+2)^2 = 0
Since the square of any real number is always non-negative, the equation (x+2)^2 = 0 has only one solution:
x + 2 = 0
x = -2
So, we found a critical point at x = -2.
Now, let's analyze the behavior of f(x) around the critical point and compare it to the given point (-2, -4):
For values of x less than -2, let's pick x = -3 as an example:
f(-3) = ((-3)+2)^3 - 4 = (-1)^3 - 4 = -1 - 4 = -5
For values of x greater than -2, let's pick x = -1 as an example:
f(-1) = ((-1)+2)^3 - 4 = (1)^3 - 4 = 1 - 4 = -3
So, we can see that for the function f(x) = (x+2)^3 - 4, the value of the function at x = -2 is f(-2) = ((-2)+2)^3 - 4 = 0 - 4 = -4. This matches the given point (-2, -4).
Therefore, the given point (-2, -4) is a critical point that is not an extremum. So, the correct answer is C) A critical point but not an extremum.
You are correct!