Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problema 116

Capitolo 4, Problema 116

Concavity of parabolas Consider the general parabola described by the function f(x) = ax² + bx + c. For what values of a, b, and c is f concave up? For what values of a, b, and c is f concave down?

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To determine the concavity of the parabola described by f(x) = ax² + bx + c, we need to examine the second derivative of the function. The second derivative provides information about the curvature of the graph.

First, find the first derivative of f(x). The first derivative, f'(x), is obtained by differentiating f(x) with respect to x: f'(x) = 2ax + b.

Next, find the second derivative, f''(x), by differentiating f'(x) with respect to x: f''(x) = 2a.

The concavity of the parabola is determined by the sign of the second derivative, f''(x). If f''(x) > 0, the parabola is concave up. If f''(x) < 0, the parabola is concave down.

Since f''(x) = 2a, the parabola is concave up when a > 0 and concave down when a < 0. The values of b and c do not affect the concavity, as they do not appear in the second derivative.

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Concavity

Concavity refers to the direction in which a curve bends. A function is concave up if its graph opens upwards, resembling a cup, and concave down if it opens downwards, resembling a cap. This behavior is determined by the second derivative of the function; if the second derivative is positive, the function is concave up, and if it is negative, the function is concave down.

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Determining Concavity Given a Function

Second Derivative Test

The second derivative test is a method used to determine the concavity of a function. For a function f(x), the second derivative, denoted as f''(x), provides information about the curvature of the graph. If f''(x) > 0 for all x in an interval, the function is concave up on that interval; if f''(x) < 0, it is concave down. This test is crucial for analyzing the behavior of polynomial functions like parabolas.

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Quadratic Function

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants. The coefficient 'a' determines the direction of the parabola: if a > 0, the parabola opens upwards (concave up), and if a < 0, it opens downwards (concave down). Understanding the role of 'a' is essential for determining the concavity of the parabola.

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Introduction to Polynomial Functions

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