91Ó°ÊÓ

The second derivative of a function provides crucial information about the nature of its graph, particularly in identifying inflection points. Inflection points occur where the concavity of the function changes, meaning the shape of the graph shifts from being curved upwards (concave up) to downwards (concave down), or vice versa.
To compute the second derivative, we must first find the first derivative of the given function. The function given is \( f(x) = 2e^{-x^2} \). Using the chain rule, the first derivative is \( f'(x) = -4x \, e^{-x^2} \).
After obtaining the first derivative, we then apply the product rule and the chain rule again to find the second derivative:

• Differentiate \(-4x\), which is simply \(-4\).
• Find the derivative of \(e^{-x^2}\), using the chain rule, resulting in \(-2x \, e^{-x^2}\).

This gives us the second derivative: \( f''(x) = -4e^{-x^2} + 4x^2 e^{-x^2} \). Setting the second derivative equal to zero helps locate potential inflection points.

The chain rule is a fundamental tool in calculus used for derivating composite functions. When a function is embedded inside another, the chain rule becomes necessary to differentiate it correctly.
For example, in the function \( f(x) = 2e^{-x^2} \), the expression \(-x^2\) is embedded within the exponential function \(e^{u}\) where \( u = -x^2 \). To differentiate \(e^{-x^2}\), we employ the chain rule:

• Take the derivative of the outer function \( e^u \), which is \( e^u \) itself.
• Multiply it by the derivative of the inner function \( u = -x^2 \), resulting in \(-2x\).

Hence, we obtain \( (e^{-x^2})' = -2x \, e^{-x^2} \). This application of the chain rule is essential for correctly finding the derivatives needed for exploring inflection points.

Concavity describes the direction in which a curve opens and plays a pivotal role in understanding the behavior of functions for various regions of their domain. Calculus defines concave up as resembling an upturned bowl (\( f''(x) > 0 \)) and concave down as a downturned bowl (\( f''(x) < 0 \)).
When we evaluate \( f''(x) \), the sign indicates the concavity:

• If \( f''(x) < 0 \) for a certain interval, the graph curves downwards in that region.
• If \( f''(x) > 0 \), it curves upwards.

To determine changes in concavity, we assess \( f''(x) \) around potential inflection points. Testing values around \( x = -1 \) and \( x = 1 \) confirmed the change in concavity, signifying genuine inflection points at \( x = -1 \) and \( x = 1 \). Recognizing these transitions is vital in understanding a function's graphical behavior.

The product rule in calculus is essential when differentiating functions that are multiplied together. For a function that is a product of two separate functions \( u(x) \) and \( v(x) \), the derivative \( (uv)' \) is given by \( u'v + uv' \).
This was particularly useful in our problem, where we needed to differentiate \(-4x \, e^{-x^2}\) to find the second derivative. Applying the product rule involves:

• Differentiating \( -4x \), which results in \(-4\).
• Differentiating \( e^{-x^2} \) using the chain rule, yielding \(-2x \, e^{-x^2}\).
• Combining these results as \(-4 \, e^{-x^2} + 4x^2 \, e^{-x^2}\).

Utilizing the product rule allows us to manage complex derivatives effectively when functions are products, a necessary step when determining inflection points and examining a function's behavior.