132. Let f(x) = e^x / (1 + e^(2x)).
b. Find all inflection points for f.

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Recall that inflection points occur where the second derivative of the function changes sign, which means we need to find where the second derivative \( f''(x) \) is zero or undefined and verify a sign change around those points.

Start by finding the first derivative \( f'(x) \) of the function \( f(x) = \frac{e^x}{1 + e^{2x}} \). Use the quotient rule: \[ f'(x) = \frac{(e^x)'(1 + e^{2x}) - e^x(1 + e^{2x})'}{(1 + e^{2x})^2} \]. Remember that \( (e^x)' = e^x \) and \( (e^{2x})' = 2e^{2x} \).

Simplify the expression for \( f'(x) \) carefully by expanding the numerator and combining like terms, then write the simplified form of \( f'(x) \).

Next, find the second derivative \( f''(x) \) by differentiating \( f'(x) \) again. Since \( f'(x) \) is a quotient, apply the quotient rule once more, or consider rewriting \( f'(x) \) in a simpler form before differentiating.

Set \( f''(x) = 0 \) and solve for \( x \). Then, check the sign of \( f''(x) \) on intervals around these points to confirm where the concavity changes, identifying the inflection points.

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Inflection Points

Inflection points are points on the graph of a function where the concavity changes from concave up to concave down or vice versa. These points occur where the second derivative of the function is zero or undefined, provided there is a change in concavity around those points.

Second Derivative

The second derivative of a function measures the rate of change of the first derivative and provides information about the concavity of the function. Calculating the second derivative and analyzing its sign helps determine where the function is concave up or down, which is essential for finding inflection points.

Exponential Functions and Their Derivatives

Exponential functions like e^x have derivatives that are proportional to themselves, making differentiation straightforward but requiring careful application of the chain and quotient rules when combined with other functions. Understanding how to differentiate expressions involving e^x and e^(2x) is crucial for finding the first and second derivatives of f(x).

132. Let f(x) = e^x / (1 + e^(2x)).b. Find all inflection points for f.