May you please find the minimum and maximum please and thank you. It would be a great help. **Problem Statement:**Find the x-value of all points where the function below has any relative extrema. Find the value(s) of any relative extrema.\[ G(x) = x^6 \, e^x - 5 \]**Question:**Select the correct choice below and, if necessary, fill in the answer box to complete your choice.- **A.** The function has a relative minimum at the point(s) $\boxed{\phantom{0}}$. (Simplify your answer. Type an ordered pair. Use integers or decimals for any numbers in the expression. Round to two decimal places as needed. Use a comma to separate answers as needed.)- **B.** The function has no relative minimum.

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Description: May you please find the minimum and maximum please and thank you. It would be a great help. **Problem Statement:**Find the x-value of all points where the function below has any relative extrema. Find the value(s) of any relative extrema.\[ G(x) = x^

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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May you please find the minimum and maximum please and thank you. It would be a great help.

**Problem Statement:**

Find the x-value of all points where the function below has any relative extrema. Find the value(s) of any relative extrema.

\[ G(x) = x^6 \, e^x - 5 \]

**Question:**

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

- **A.** The function has a relative minimum at the point(s) $\boxed{\phantom{0}}$.
(Simplify your answer. Type an ordered pair. Use integers or decimals for any numbers in the expression. Round to two decimal places as needed. Use a comma to separate answers as needed.)

- **B.** The function has no relative minimum.

Expert Solution

Step 1

Given function is: $G (x) = x^{6} e^{x} - 5$

To find extreme points, find the points where the derivative of the function is equal to 0;

Derivative of $G (x) = x^{6} e^{x} - 5$ :

$G' (x) = e^{x} [x^{6} + 6 x^{5}]$

Points where $G' (x) = 0$ are:

$G' (x) = 0 e^{x} [x^{6} + 6 x^{5}] = 0 e^{x} x^{5} [x + 6] = 0$

So, the two possibilities for x are: $x = 0 o r x = - 6$

At these two points, the function has extreme points.

Now, use second derivative test to find if the points has local minima or local maxima:

$\begin{array}{rcl} G'' (x) & = & e^{x} [(x^{6} + 6 x^{5}) + (6 x^{5} + 30 x^{4})] \\ = & e^{x} [x^{6} + 12 x^{5} + 30 x^{4}] \\ = & e^{x} x^{4} [x^{2} + 12 x + 30] \end{array}$

When second derivative at extreme point is negative, then function has local maxima at that point and if the derivative is positive, then the function has local minima at that point.

The second derivative at point $x = 0$

$G'' (0) = 0$

Therefore, the function has local minima at point $x = 0$ .

The second derivative at point $x = - 6$

$\begin{array}{rcl} G'' (- 6) & = & e^{- 6} {(- 6)}^{4} [{(- 6)}^{2} + 12 (- 6) + 30] \\ = & e^{- 6} 6^{4} [36 - 72 + 30] \\ = & e^{- 6} 6^{4} (- 6) \\ = & - e^{- 6} 6^{5} (< 0) \end{array}$

Therefore, the function has local maxima at point $x = - 6$ .

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