Final Answer:
The relative extreme points of the function f(x) = -4 + x^(2/3) occur at the critical points where the derivative equals zero or is undefined.
Step-by-step explanation:
To find the relative extreme points of the function f(x) = -4 + x^(2/3), we start by calculating its derivative, f'(x). For the given function, using the power rule for differentiation, the derivative is:
f'(x) = d/dx (-4 + x^(2/3))
f'(x) = 2/3 * x^(-1/3)
To find the critical points, we set the derivative equal to zero and solve for x:
2/3 * x^(-1/3) = 0
This derivative is undefined at x = 0, but it doesn't exist within the domain of the original function. Thus, x = 0 doesn't affect the function's extreme points.
Setting the derivative equal to zero:
2/3 * x^(-1/3) = 0
x^(-1/3) = 0
This equation doesn't yield any real solutions for x since x^(-1/3) cannot be zero. Therefore, this function, f(x) = -4 + x^(2/3), doesn't possess any relative extreme points in the real number domain.