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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.4.113
Capitolo 4, Problema 4.4.113

113. If b, c, and d are constants, for what value of b will the curve y = x^3 + bx^2 + cx + d have a
point of inflection at x = 1? Give reasons for your answer.

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To find the point of inflection, we need to determine where the second derivative of the function changes sign. Start by finding the first derivative of the function y = x^3 + bx^2 + cx + d. The first derivative is y' = 3x^2 + 2bx + c.
Next, find the second derivative of the function. Differentiate the first derivative to get y'' = 6x + 2b.
A point of inflection occurs where the second derivative is zero or undefined, and changes sign. Set the second derivative equal to zero: 6x + 2b = 0.
Solve the equation 6x + 2b = 0 for x = 1, since the point of inflection is given to be at x = 1. Substitute x = 1 into the equation: 6(1) + 2b = 0.
Solve the equation 6 + 2b = 0 for b to find the value of b that ensures the curve has a point of inflection at x = 1.

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Point of Inflection

A point of inflection on a curve is where the concavity changes from concave up to concave down, or vice versa. This occurs where the second derivative of the function changes sign. To find such points, we set the second derivative equal to zero and solve for the variable.
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Critical Points

Second Derivative

The second derivative of a function, denoted as f''(x), provides information about the concavity of the function. It is the derivative of the first derivative, f'(x). If f''(x) > 0, the function is concave up, and if f''(x) < 0, it is concave down. A change in sign of f''(x) indicates a potential point of inflection.
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The Second Derivative Test: Finding Local Extrema

Polynomial Derivatives

To find the derivatives of a polynomial function, apply the power rule: for a term ax^n, the derivative is n*ax^(n-1). For the function y = x^3 + bx^2 + cx + d, the first derivative is 3x^2 + 2bx + c, and the second derivative is 6x + 2b. These derivatives are used to analyze the behavior of the curve, such as finding points of inflection.
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Introduction to Polynomial Functions
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Root Finding

2. Use Newton's method to estimate the one real solution of x^3 +3x + 1 = 0. Start with x_0 = 0 and then find x_2.

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Checking the Mean Value Theorem


Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.


f(x) = {2x − 3, 0 ≤ x ≤ 2

6x − x² − 7, 2 < x ≤ 3

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Finding Extrema from Graphs


In Exercises 11–14, match the table with a graph.


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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.

82. y' = sin t, for 0 ≤ t ≤ 2π

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99. In Exercises 99 and 100, the graph of f' is given. Determine x-values corresponding to inflection points for the graph of f.

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Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(2x³ − 5x + 7) dx

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