Questions: Let f(x)=e^(-5 x^2) a. Compute f''(x). f''(x)= b. Find the x-value(s) where f''(x)=0, and then list them (separated by commas) in the box below. If there are none, enter DNE. x=

Let f(x)=e^(-5 x^2) a. Compute f''(x). f''(x)= b. Find the x-value(s) where f''(x)=0, and then list them (separated by commas) in the box below. If there are none, enter DNE. x=
Transcript text: Let $f(x)=e^{-5 x^{2}}$ a. Compute $f^{\prime \prime}(x)$. \[ f^{\prime \prime}(x)= \] $\square$ b. Find the $x$-value(s) where $f^{\prime \prime}(x)=0$, and then list them (separated by commas) in the box below. If there are none, enter DNE. \[ x= \] $\square$
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Solution

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Solution Steps

To solve the given problem, we need to perform the following steps:

a. Compute the second derivative of the function \( f(x) = e^{-5x^2} \). This involves differentiating the function twice with respect to \( x \).

b. Find the \( x \)-value(s) where the second derivative \( f''(x) = 0 \). This requires solving the equation obtained from the second derivative for \( x \).

Step 1: Compute the Second Derivative

We start with the function \( f(x) = e^{-5x^2} \). The first derivative is calculated as follows: \[ f'(x) = -10x e^{-5x^2} \] Next, we differentiate \( f'(x) \) to find the second derivative: \[ f''(x) = 100x^2 e^{-5x^2} - 10 e^{-5x^2} \]

Step 2: Set the Second Derivative to Zero

To find the \( x \)-value(s) where \( f''(x) = 0 \), we set the second derivative equal to zero: \[ 100x^2 e^{-5x^2} - 10 e^{-5x^2} = 0 \] Factoring out \( e^{-5x^2} \) (which is never zero), we have: \[ 100x^2 - 10 = 0 \] Solving for \( x^2 \): \[ 100x^2 = 10 \implies x^2 = \frac{1}{10} \implies x = \pm \frac{\sqrt{10}}{10} \]

Final Answer

The second derivative is given by: \[ f''(x) = 100x^2 e^{-5x^2} - 10 e^{-5x^2} \] The \( x \)-values where \( f''(x) = 0 \) are: \[ x = -\frac{\sqrt{10}}{10}, \frac{\sqrt{10}}{10} \] Thus, the final answers are: \[ \boxed{f''(x) = 100x^2 e^{-5x^2} - 10 e^{-5x^2}} \] \[ \boxed{x = -\frac{\sqrt{10}}{10}, \frac{\sqrt{10}}{10}} \]

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