Each of Exercises 43–48 gives the first derivative of a function y = ƒ(𝓍). (a) At what points, if any, does the graph of ƒ have a local maximum, local minimum, or inflection point? (b) Sketch the general shape of the graph.
y' = 𝓍² ― 𝓍―6

검증된 단계별 안내

First, identify the critical points by setting the first derivative y' = 𝓍² - 𝓍 - 6 equal to zero and solving for 𝓍. This will help us find potential points for local maxima, minima, or inflection points.

Factor the quadratic equation 𝓍² - 𝓍 - 6 = 0 to find the values of 𝓍. The equation can be factored as (𝓍 - 3)(𝓍 + 2) = 0, giving the critical points 𝓍 = 3 and 𝓍 = -2.

To determine whether these critical points are local maxima or minima, use the second derivative test. Compute the second derivative y'' by differentiating y' = 𝓍² - 𝓍 - 6 again, resulting in y'' = 2𝓍 - 1.

Evaluate the second derivative at the critical points: y''(3) and y''(-2). If y''(𝓍) > 0, the point is a local minimum; if y''(𝓍) < 0, the point is a local maximum. If y''(𝓍) = 0, further analysis is needed to determine the nature of the point.

To find inflection points, set the second derivative y'' = 2𝓍 - 1 equal to zero and solve for 𝓍. This will indicate where the concavity of the function changes, which is a characteristic of inflection points.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

First Derivative Test

The First Derivative Test is a method used to determine local extrema of a function. By analyzing the sign of the first derivative, we can identify intervals where the function is increasing or decreasing. A local maximum occurs where the derivative changes from positive to negative, while a local minimum occurs where it changes from negative to positive.

Critical Points

Critical points are values of x where the first derivative is either zero or undefined. These points are essential for finding local maxima, minima, and inflection points. In the given derivative, we find critical points by solving the equation y' = 0, which helps us identify where the function's behavior changes.

Inflection Points

Inflection points occur where the concavity of a function changes, which can be determined by analyzing the second derivative. However, in the context of the first derivative, inflection points can also be inferred from changes in the behavior of the first derivative itself. Identifying these points helps in sketching the overall shape of the graph.

Each of Exercises 43–48 gives the first derivative of a function y = ƒ(𝓍). (a) At what points, if any, does the graph of ƒ have a local maximum, local minimum, or inflection point? (b) Sketch the general shape of the graph.y' = 𝓍² ― 𝓍―6