Final answer:
To find the extremum points of the function f(x)=x²/³ in the interval [−1,2], we examine its first and second derivatives. We find that the function has a potential maximum at x=0 and no minima points within the interval.
Step-by-step explanation:
The student is asking to find the points at which the function f(x)=x²/³ has a maximum or a minimum value in the interval [−1,2]. To identify the extremum points, we must investigate the function's first and second derivative within this interval. The first derivative, f'(x), represents the slope of the function and is used to identify potential extrema. Where f'(x)=0, we have a potential maximum or minimum. We then use the second derivative, f"(x), to determine the nature of these extreme points; a positive second derivative indicates a minimum, while a negative second derivative indicates a maximum.
First, we find the first derivative of f(x): f'(x)=⅔x-1/⅓. Setting this equal to zero, we solve for x. However, since the derivative does not exist at x=0, we include the endpoints of the interval. The second derivative is f"(x) = −2/9x-4/3; which is negative throughout the interval (excluding x=0 where it is not defined), suggesting that there are no minimum points and thus only one potential maximum at x = 0, since the endpoints are not extremes.
Thus, we conclude that at x=0, f(x) has a potential maximum, given the behavior of the function and its derivatives within the given interval.