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

Guida verificata passo dopo passo

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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