Consider the function f(x)=x e^-x^2 / 2. Note: \ds limx →±∞ f(x)=0. 1. Find all inflection point

Name: Problem 7, Chapter 3: CLP-1 Differential Calculus 1
Uploaded: 2022-02-23T21:26:08-08:00
Duration: 1 min 35 s
Channel: Sriparna Bhattacharjee
Description: Consider the function f(x)=x e^-x^2 / 2. Note: \ds limx →±∞ f(x)=0. 1. Find all inflection points and intervals of increase, decrease, convexity up, and convexity down. You may use without proof the formula f^''(x)=(x^3-3 x) e^-x^2 / 2. 2. Find local and global minima and maxima. 3. Use all the above to draw a graph for f. Indicate all special points on the graph.

step 1 Solution A is FX is equal to E to x minus x cubed, F double dashed X is equal to E to x minus 6x

Question

Consider the function $f(x)=x e^{-x^2 / 2}$. Note: $\backslash \mathrm{ds} \lim _{x \rightarrow \pm \infty} f(x)=0$. 1. Find all inflection points and intervals of increase, decrease, convexity up, and convexity down. You may use without proof the formula $f^{\prime \prime}(x)=\left(x^3-3 x\right) e^{-x^2 / 2}$. 2. Find local and global minima and maxima. 3. Use all the above to draw a graph for $f$. Indicate all special points on the graph.

   Consider the function $f(x)=x e^{-x^2 / 2}$.
Note: $\backslash \mathrm{ds} \lim _{x \rightarrow \pm \infty} f(x)=0$.
1. Find all inflection points and intervals of increase, decrease, convexity up, and convexity down. You may use without proof the formula $f^{\prime \prime}(x)=\left(x^3-3 x\right) e^{-x^2 / 2}$.
2. Find local and global minima and maxima.
3. Use all the above to draw a graph for $f$. Indicate all special points on the graph.

CLP-1 Differential Calculus 1

Joel Feldman, Andrew… 1st Edition

Chapter 3, Problem 7

Instant Answer

Solved by Expert Sriparna Bhattacharjee

02/23/2022

Step 1

The first derivative is given by the product rule: \[ f'(x) = \frac{d}{dx} \left( x e^{-x^2 / 2} \right) = e^{-x^2 / 2} + x \cdot \frac{d}{dx} \left( e^{-x^2 / 2} \right). \] The derivative of $ e^{-x^2 / 2} $ is: \[ \frac{d}{dx} \left( e^{-x^2 / 2} \right) = Show more…

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Consider the function $f(x)=x e^{-x^2 / 2}$. Note: $\backslash \mathrm{ds} \lim _{x \rightarrow \pm \infty} f(x)=0$. 1. Find all inflection points and intervals of increase, decrease, convexity up, and convexity down. You may use without proof the formula $f^{\prime \prime}(x)=\left(x^3-3 x\right) e^{-x^2 / 2}$. 2. Find local and global minima and maxima. 3. Use all the above to draw a graph for $f$. Indicate all special points on the graph.