### Analysis of the Function $ g(x) = x^4 - 2x^4 $Consider the following problems related to the function:a) **Find the inflection points of $ g(x) $** An inflection point is where the concavity of the function changes. To find them, calculate the second derivative of $ g(x) $ and solve for when it equals zero.b) **Find the intervals where $ g $ is concave up and concave down** A function is concave up where its second derivative is positive and concave down where it is negative. Determine these intervals by analyzing the sign of the second derivative.c) **Find the extrema of $ g(x) $ using the second derivative test** The second derivative test helps determine whether a critical point is a local minimum, maximum, or a saddle point. Calculate the first derivative to find critical points, then use the second derivative to classify them.### Graphs and Diagrams- **Graphs:** Visualizing $ g(x) $ and its derivatives can aid understanding. Graphs typically show plots of $ g(x) $ and its second derivative to illustrate regions of concavity and inflection points.- **Diagrams:** No additional diagrams were provided in the text.Ensure to solve the derivatives accurately and analyze the results systematically for proper interpretation.

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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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### Analysis of the Function $ g(x) = x^4 - 2x^4 $

Consider the following problems related to the function:

a) **Find the inflection points of $ g(x) $**
An inflection point is where the concavity of the function changes. To find them, calculate the second derivative of $ g(x) $ and solve for when it equals zero.

b) **Find the intervals where $ g $ is concave up and concave down**
A function is concave up where its second derivative is positive and concave down where it is negative. Determine these intervals by analyzing the sign of the second derivative.

c) **Find the extrema of $ g(x) $ using the second derivative test**
The second derivative test helps determine whether a critical point is a local minimum, maximum, or a saddle point. Calculate the first derivative to find critical points, then use the second derivative to classify them.

### Graphs and Diagrams

- **Graphs:** Visualizing $ g(x) $ and its derivatives can aid understanding. Graphs typically show plots of $ g(x) $ and its second derivative to illustrate regions of concavity and inflection points.
- **Diagrams:** No additional diagrams were provided in the text.

Ensure to solve the derivatives accurately and analyze the results systematically for proper interpretation.

Expert Solution

Step 1

$G i v e n, f (x) = x^{4} - 2 x^{4} S i m p l i f y i n g f (x) w e g e t, f (x) = - x^{4}$

Step 2

$I n f l e c t i o n P o i n t s A n i n f l e c t i o n p o i n t i s a p o i n t o n t h e g r a p h a t w h i c h t h e s e c o n d d e r i v a t i v e c h a n g e s s i g n If f'' (x) > 0 then f (x) concave upwards . If f'' (x) < 0 then f (x) concave downwards . f'' (x) = - 12 x^{2} P u t f'' (x) = 0 - 12 x^{2} = 0 x = 0 D o m a i n o f - x^{4} i s - \infty < x < \infty A l l i n f l e c t i o n p o i n t s a r e i n d o m a i n a n d n o t o n d o m a i n e d g e s x = 0 Combine inflection point (s) : x = 0 With the domain to get f'' (x) sign intervals : - \infty < x < 0, 0 < x < \infty S u m m a r y o f t h e s i g n i n t e r v a l s b e h a v i o r : \begin{matrix} - \infty < x < 0 & x = 0 \\ S i g n & - & 0 \\ B e h a v i o r & C o n c a v e D o w n w a r d & N A \end{matrix} \begin{matrix} 0 < x < \infty \\ - \\ C o n c a v e D o w n w a r d \end{matrix} T h u s, T h e r e i s n o i n f l e c t i o n p o i n t s .$

Step 3

$T o f i n d e x t r e m a : Suppose that x = c is a critical point of f' (c) such that f' (c) = 0 and that f'' (x) is continuous in a region around x = c . Then, If f'' (c) < 0 then x = c is a local maximum . If f'' (c) > 0 then x = c is a local minimum . If f'' (c) = 0 then test failed and x = c can be a local maximum, local minimum or neither . C r i t i c a l P o i n t s : x = 0 f'' (x) = - 12 x^{2} Check the sign of f'' (x) = - 12 x^{2} at each critical point Since f'' (x) = 0 at x = 0, the second derivative test is inconclusive$

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