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For

x \in \mathbb{R}

, find the maximum of: \[ \frac{(1+x)^{8}+16 x^{4}}{\left...

Sep 5, 2024

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Solution by Steps

step 1

We start with the function to maximize:

f(x) = \frac{(1+x)^{8}+16 x^{4}}{(1+x^{2})^{4}}

step 2

We can simplify the expression to

f(x) = \frac{16 x^{4} + (x + 1)^{8}}{(x^{2} + 1)^{4}}

step 3

The function can be analyzed by breaking it down into its components, leading to

\frac{(x + 1)^{8}}{(x^{2} + 1)^{4}} + \frac{16 x^{4}}{(x^{2} + 1)^{4}}

step 4

The maximum value occurs when

x

is evaluated at critical points derived from the first derivative of

f(x)

set to zero

step 5

After evaluating the critical points, we find that the maximum value of the function is

16

x = 1

Answer

The maximum value of the function is

16

x = 1

Key Concept

Finding the maximum of a rational function involves analyzing its critical points and evaluating the function at those points.

Explanation

The maximum value is determined by simplifying the function and finding where its derivative equals zero, leading to the maximum value of

16

x = 1