# What are the points of inflection, if any, of #f(x)=2e^(-x^2) #?

Thus, we determine:

That is:

plot{2e^(-x^2) [-2.54, 2.46, -2.26, 0.24]}

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To find the points of inflection, we need to find the second derivative of the function.

The second derivative of f(x) = 2e^(-x^2) is: f''(x) = 4x^2 e^(-x^2) - 2e^(-x^2).

Next, we set the second derivative equal to zero and solve for x: 4x^2 e^(-x^2) - 2e^(-x^2) = 0.

Factoring out e^(-x^2), we get: e^(-x^2)(4x^2 - 2) = 0.

This equation equals zero when e^(-x^2) = 0 or when 4x^2 - 2 = 0.

However, e^(-x^2) is never zero, so we only need to solve 4x^2 - 2 = 0.

Solving for x gives x = ±√(2/4) = ±√(1/2) = ±(1/√2).

Therefore, the points of inflection are at (±(1/√2), f(1/√2)).

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