**Problem Statement:**Sketch the graph of the function $ y = xe^{-21x^2} $. Indicate the transition points (local extrema and points of inflection).(Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list of $ x $-coordinates. Enter NULL in answer field if there is no such point.)**Solutions:**- Local maximum at $ x = \frac{1}{\sqrt{42}} $- Local minimum at $ x = -\frac{1}{\sqrt{42}} $- Inflection at $ x = \frac{1}{\sqrt{14}} \times e^{3/2} $ [Help (fractions)]

Answered: Sketch the graph of the function y =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Concept explainers

Question

**Problem Statement:**

Sketch the graph of the function $ y = xe^{-21x^2} $. Indicate the transition points (local extrema and points of inflection).

(Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list of $ x $-coordinates. Enter NULL in answer field if there is no such point.)

**Solutions:**

- Local maximum at $ x = \frac{1}{\sqrt{42}} $
- Local minimum at $ x = -\frac{1}{\sqrt{42}} $
- Inflection at $ x = \frac{1}{\sqrt{14}} \times e^{3/2} $

[Help (fractions)]

Expert Solution

Step 1

Here the given function is $y = x e^{- 21 x^{2}}$

Step 2

Find the derivative of the function using product rule of derivatives.

$\begin{array}{rcl} y' & = & e^{- 21 x^{2}} \frac{d}{d x} (x) + x \frac{d}{d x} (e^{- 21 x^{2}}) \\ = & e^{- 21 x^{2}} (1) + x e^{- 21 x^{2}} \frac{d}{d x} (- 21 x^{2}) \\ = & e^{- 21 x^{2}} + x e^{- 21 x^{2}} (- 42 x) \\ = & e^{- 21 x^{2}} (1 - 42 x^{2}) \end{array}$

Step 3

Find the critical values by equating y' to 0.

$y' = 0 \begin{array}{rcl} e \end{array} \begin{array}{rcl} ^{- 21 x^{2}} \end{array} \begin{array}{rcl} (1 - 42 x^{2}) \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} 0 \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} 1 \end{array} \begin{array}{rcl} - \end{array} \begin{array}{rcl} 42 \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} ^{2} \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} 0 \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} \frac{1}{\sqrt{42}} \end{array} \begin{array}{rcl} , \end{array} \begin{array}{rcl} - \end{array} \begin{array}{rcl} \frac{1}{\sqrt{42}} \end{array}$

Step 4

Find the second derivative of the function.

$\begin{array}{rcl} y'' & = & e^{- 21 x^{2}} \frac{d}{d x} (1 - 42 x^{2}) + (1 - 42 x^{2}) \frac{d}{d x} (e^{- 21 x^{2}}) \\ = & e^{- 21 x^{2}} (- 84 x) + (1 - 42 x^{2}) e^{- 21 x^{2}} \frac{d}{d x} (- 21 x^{2}) \\ = & - 84 x e^{- 21 x^{2}} + (1 - 42 x^{2}) e^{- 21 x^{2}} (- 42 x) \\ = & x e^{- 21 x^{2}} [- 84 - 42 + 1764 x^{2}] \\ = & x e^{- 21 x^{2}} [- 126 + 1764 x^{2}] \end{array}$

Step by step

Solved in 7 steps with 1 images

Knowledge Booster

$Application of Differentiation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions