Second Derivative Sign Analysis

00:00 3 marks

Given the function \(f(x)\) with \(f”(x)=\frac{(x-2)(x+3)}{x^2+1}\), construct a sign chart for \(f”(x)\) and identify the potential inflection points. Explain how the changes in the sign of \(f”(x)\) indicate changes in the concavity of \(f(x)\).

Hint Hint

Hint

Determine the zeros of the numerator of \(f”(x)\) to find critical points. Then, test the sign of \(f”(x)\) in each interval defined by these points to complete the sign chart.

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Answer Answer

Complete Answer

Potential inflection points occur at \(x = -3\) and \(x = 2\), as the sign of \(f”(x)\) changes at these points, indicating changes in concavity.

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