Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.


ƒ(x) = x²/₃(4 -x²) on -2,2

Verified step by step guidance

1. First, understand that absolute extrema refer to the highest and lowest values of a function on a given interval. We need to find these values for ƒ(x) = x^(2/3)(4 - x^2) on the interval [-2, 2].
2. To find the extrema, start by finding the derivative of the function ƒ(x). Use the product rule for differentiation, which states that if you have a function that is the product of two functions, u(x) and v(x), then its derivative is u'(x)v(x) + u(x)v'(x).
3. Set the derivative equal to zero to find critical points. Critical points are where the derivative is zero or undefined, and they are potential candidates for extrema. Solve the equation to find the values of x that make the derivative zero.
4. Evaluate the function ƒ(x) at the critical points found in the previous step, as well as at the endpoints of the interval, x = -2 and x = 2. This will give you the values of the function at these points.
5. Compare the values obtained from evaluating the function at the critical points and endpoints. The largest value will be the absolute maximum, and the smallest value will be the absolute minimum on the interval [-2, 2].

Key Concepts

Absolute Extrema

Absolute extrema refer to the highest and lowest values of a function over a specified interval. To find these values, one must evaluate the function at critical points, where the derivative is zero or undefined, as well as at the endpoints of the interval. The largest of these values is the absolute maximum, while the smallest is the absolute minimum.

Critical Points

Critical points are values of the independent variable where the derivative of the function is either zero or does not exist. These points are essential in determining the behavior of the function, as they can indicate potential locations for local maxima or minima. To find critical points, one must first compute the derivative of the function and solve for when it equals zero or is undefined.

Endpoints of the Interval

Endpoints of the interval are the boundary values of the domain over which the function is being analyzed. In the context of finding absolute extrema, it is crucial to evaluate the function at these endpoints, as they can potentially yield the highest or lowest values of the function. For the given interval [-2, 2], the function must be evaluated at both x = -2 and x = 2.