Determine the intervals on which the function g(𝓍) = ∫ₓ⁰ t / (t² + 1) dt is concave up or concave down.

검증된 단계별 안내

Step 1: Recognize that the function g(𝓍) is defined as a definite integral with a variable upper limit. This means g(𝓍) is a function whose derivative can be found using the Fundamental Theorem of Calculus. Specifically, g'(𝓍) = 𝓍 / (𝓍² + 1).

Step 2: To determine concavity, calculate the second derivative g''(𝓍). Differentiate g'(𝓍) = 𝓍 / (𝓍² + 1) using the quotient rule: g''(𝓍) = [(𝓍² + 1)(1) - 𝓍(2𝓍)] / (𝓍² + 1)².

Step 3: Simplify g''(𝓍). The numerator becomes (𝓍² + 1) - 2𝓍² = 1 - 𝓍². Thus, g''(𝓍) = (1 - 𝓍²) / (𝓍² + 1)².

Step 4: Analyze the sign of g''(𝓍) to determine concavity. The numerator (1 - 𝓍²) is positive when 𝓍² < 1 (i.e., -1 < 𝓍 < 1) and negative when 𝓍² > 1 (i.e., 𝓍 < -1 or 𝓍 > 1). The denominator (𝓍² + 1)² is always positive.

Step 5: Conclude that g(𝓍) is concave up on the interval (-1, 1) where g''(𝓍) > 0, and concave down on the intervals (-∞, -1) and (1, ∞) where g''(𝓍) < 0.

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주요 개념

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Concavity

Concavity refers to the direction in which a function curves. A function is concave up on an interval if its second derivative is positive, indicating that the slope of the tangent line is increasing. Conversely, it is concave down if the second derivative is negative, meaning the slope is decreasing. Understanding concavity helps in analyzing the behavior of functions and their graphs.

Second Derivative Test

The second derivative test is a method used to determine the concavity of a function. By taking the second derivative of a function, we can assess where the function is concave up or down. If the second derivative is positive at a point, the function is concave up; if negative, it is concave down. This test is crucial for identifying intervals of concavity in the given function.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, stating that if a function is defined as an integral, its derivative can be found using the integrand evaluated at the upper limit. In this case, the function g(x) is defined as an integral, and understanding how to differentiate it will allow us to find the first and second derivatives necessary for analyzing concavity.