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) = 3x⁵ - 25x³ + 60x on [-2,3]

검증된 단계별 안내

First, find the critical points of the function ƒ(x) = 3x⁵ - 25x³ + 60x by taking its derivative and setting it equal to zero. The derivative is ƒ'(x) = 15x⁴ - 75x² + 60.

Solve the equation 15x⁴ - 75x² + 60 = 0 to find the critical points. This can be done by factoring or using the quadratic formula on the resulting polynomial.

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 = 3.

Compare the values of ƒ(x) at the critical points and the endpoints to determine which is the absolute maximum and which is the absolute minimum on the interval [-2, 3].

Conclude by stating the location (x-values) and the corresponding values of the absolute maximum and minimum of the function on the given interval.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

12m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Critical Points

Critical points are values of x in the domain of a function where the derivative is either zero or undefined. These points are essential for finding local maxima and minima, as they indicate where the function's slope changes. To locate absolute extrema on a closed interval, one must evaluate the function at these critical points as well as at the endpoints of the interval.

First Derivative Test

The First Derivative Test is a method used to determine whether a critical point is a local maximum, local minimum, or neither. By analyzing the sign of the derivative before and after the critical point, one can infer the behavior of the function. If the derivative changes from positive to negative, the critical point is a local maximum; if it changes from negative to positive, it is a local minimum.

Evaluating Endpoints

When finding absolute extrema on a closed interval, it is crucial to evaluate the function at the endpoints of the interval in addition to the critical points. The absolute maximum or minimum could occur at these endpoints, especially if the function is not continuous or has significant changes in behavior at the edges of the interval. Thus, comparing the function values at critical points and endpoints ensures all potential extrema are considered.

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) = 3x⁵ - 25x³ + 60x on [-2,3]