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.
74. y' = (x² - 2x)(x - 5)²

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Step 1: Start by finding the second derivative y''. To do this, apply the product rule to y' = (x² - 2x)(x - 5)². Recall the product rule: if y' = u(x)v(x), then y'' = u'(x)v(x) + u(x)v'(x). Here, u(x) = (x² - 2x) and v(x) = (x - 5)².

Step 2: Compute the derivative of u(x) = (x² - 2x). Use the power rule to find u'(x) = 2x - 2.

Step 3: Compute the derivative of v(x) = (x - 5)². Use the chain rule to find v'(x) = 2(x - 5).

Step 4: Substitute u(x), u'(x), v(x), and v'(x) into the product rule formula for y''. This gives y'' = [(2x - 2)(x - 5)²] + [(x² - 2x)(2(x - 5))]. Simplify this expression by expanding and combining like terms.

Step 5: Use the second derivative y'' to analyze the concavity of the graph of f(x). Identify where y'' > 0 (indicating concave up) and y'' < 0 (indicating concave down). Combine this information with the critical points from y' = 0 to sketch the general shape of the graph of f(x).

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주요 개념

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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 or minima.

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, and if y'' is negative, the graph is concave down. This information is crucial for sketching the general shape of the function.

Graphing Procedure Steps

The graphing procedure involves several steps to analyze the behavior of a function based on its derivatives. Steps typically include finding critical points, determining intervals of increase and decrease, analyzing concavity using the second derivative, and identifying points of inflection. These steps collectively help in sketching an accurate representation of the function's graph.

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.74. y' = (x² - 2x)(x - 5)²