Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.
71. y' = x(x² - 12)

검증된 단계별 안내

First, find the second derivative y'' by differentiating y' = x(x² - 12). Use the product rule for differentiation, which states that if you have a function u(x)v(x), its derivative is u'(x)v(x) + u(x)v'(x). Here, let u(x) = x and v(x) = x² - 12.

Differentiate u(x) = x to get u'(x) = 1. Differentiate v(x) = x² - 12 to get v'(x) = 2x.

Apply the product rule: y'' = u'(x)v(x) + u(x)v'(x) = 1(x² - 12) + x(2x). Simplify this expression to find y''.

Now, analyze the critical points and concavity using y'' to understand the general shape of the graph of f(x). Critical points occur where y'' = 0 or is undefined. Determine these points and test intervals around them to see if y'' is positive (indicating concave up) or negative (indicating concave down).

Use the information from the second derivative test to sketch the general shape of the graph of f(x). Consider the behavior at critical points and the concavity to draw a rough sketch of the graph, showing where it is increasing, decreasing, concave up, and concave down.

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

The first derivative of a function, denoted as y' or f'(x), represents the rate of change of the function with respect to x. It provides information about the slope of the tangent line to the graph of the function at any given point. Analyzing the first derivative helps identify critical points, where the function may have local maxima, minima, or points of inflection.

Second Derivative

The second derivative, denoted as y'' or f''(x), is the derivative of the first derivative. It indicates the rate of change of the slope of the function, providing insights into the concavity of the graph. If y'' is positive, the graph is concave up, suggesting a local minimum; if y'' is negative, the graph is concave down, indicating a local maximum.

Graphing Procedure

The graphing procedure involves several steps to sketch the function based on its derivatives. Steps typically include finding critical points from the first derivative, determining concavity and inflection points from the second derivative, and analyzing the behavior of the function at these points. This systematic approach helps create an accurate representation of the function's overall shape.

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.71. y' = x(x² - 12)