Points of inflection Find the x-coordinate of the point(s) of inflection of f(x) = tanh² x.

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Recall that points of inflection occur where the second derivative of the function changes sign, which typically happens where the second derivative is zero or undefined.

Start by finding the first derivative of the function \(f(x) = \tanh^2 x\). Use the chain rule: if \(f(x) = (g(x))^2\), then \(f'(x) = 2 g(x) g'(x)\). Here, \(g(x) = \tanh x\).

Calculate \(g'(x)\), the derivative of \(\tanh x\). Recall that \(\frac{d}{dx} \tanh x = \operatorname{sech}^2 x\).

Combine these results to write the first derivative: \(f'(x) = 2 \tanh x \cdot \operatorname{sech}^2 x\).

Next, find the second derivative \(f''(x)\) by differentiating \(f'(x)\). Use the product rule on \(2 \tanh x \cdot \operatorname{sech}^2 x\), then set \(f''(x) = 0\) and solve for \(x\) to find potential inflection points.

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10m

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Points of Inflection

Points of inflection occur where the concavity of a function changes, which means the second derivative changes sign. To find these points, we identify where the second derivative is zero or undefined and verify a sign change around those points.

Second Derivative

The second derivative of a function measures the curvature or concavity of the graph. It is found by differentiating the first derivative. Analyzing the second derivative helps determine where the function is concave up or down and locate inflection points.

Derivative of Hyperbolic Functions

Hyperbolic functions like tanh(x) have specific derivatives: the derivative of tanh(x) is sech²(x). Understanding these derivatives is essential for differentiating f(x) = tanh²(x) correctly and finding the first and second derivatives needed for inflection points.