Let's solve the given equation step-by-step.
The equation to solve is:
[tex]\[ 4^x + \frac{1}{4x} = 16 \cdot \frac{1}{16} \][/tex]
First, simplify the right-hand side of the equation:
[tex]\[ 16 \cdot \frac{1}{16} = 1 \][/tex]
So, the equation now becomes:
[tex]\[ 4^x + \frac{1}{4x} = 1 \][/tex]
Next, we need to solve for [tex]\( x \)[/tex].
1. We introduce substitution to make the equation easier to manage. Let [tex]\( y = 4^x \)[/tex]. Therefore, the equation becomes:
[tex]\[ y + \frac{1}{4x} = 1 \][/tex]
2. Notice that [tex]\( y = 4^x \)[/tex]. We can rewrite [tex]\( 4^x \)[/tex] as [tex]\( (2^2)^x = 2^{2x} \)[/tex], so we can let [tex]\( y = 2^{2x} \)[/tex].
3. Now, observe that in the term [tex]\(\frac{1}{4x}\)[/tex], there is an `x` in the denominator, which makes direct solving complicated. To simplify further, consider the whole equation in the context of possible values for [tex]\( x \)[/tex].
For simplicity, let’s trial some basic values of [tex]\( x \)[/tex] to find valid solutions:
### Trial for [tex]\( x = 0 \)[/tex]:
[tex]\[ 4^0 + \frac{1}{4 \cdot 0} \][/tex]
Which simplifies to:
[tex]\[ 1 + \frac{1}{0} \][/tex]
This is undefined due to division by zero.
### Trial for [tex]\( x = 1 \)[/tex]:
[tex]\[ 4^1 + \frac{1}{4 \cdot 1} \][/tex]
Which simplifies to:
[tex]\[ 4 + \frac{1}{4} \][/tex]
[tex]\[ 4 + 0.25 = 4.25 \][/tex]
Since [tex]\( 4.25 \neq 1 \)[/tex], [tex]\( x = 1 \)[/tex] is not a solution.
### Trial for [tex]\( x = -1/2 \)[/tex]:
[tex]\[ 4^{-1/2} + \frac{1}{4 \cdot (-\frac{1}{2})} \][/tex]
Which simplifies to:
[tex]\[ \left(\frac{1}{4^{1/2}}\right) + \frac{1}{-2} \][/tex]
[tex]\[ \left(\frac{1}{2}\right) + (-\frac{1}{2}) \][/tex]
[tex]\[ 0 \neq 1 \][/tex]
Since [tex]\( 0 \neq 1 \)[/tex], [tex]\( x = -1/2 \)[/tex] is not a solution.
By examining simple trials:
### Trial for [tex]\( x = 1/2 \)[/tex]:
[tex]\[ 4^{1/2} + \frac{1}{4 \cdot \frac{1}{2}} \][/tex]
Which simplifies to:
[tex]\[ 2 + \frac{1}{2} \][/tex]
[tex]\[ 2 + 0.5 = 2.5 \][/tex]
Since [tex]\( 2.5 \neq 1 \)[/tex], [tex]\( x = 1/2 \)[/tex] is not a solution either.
Given the complexity and continuous nature of the function, it's clear that the equation requires more intricate steps or specialized numerical methods to find potential boundaries/undefined calculations/non-trivial algebraic solutions. The above manual trials guide us that basic algebraic substitutive attempts don't directly solve it practically.
Hence, from manual inspection, there's no simple solution satisfying:
[tex]\[ 4^x + \frac{1}{4x} = 1 \][/tex] at trialed or trivial checkpoints. Advanced numerical algebra solutions or graphing should be recommended in such non-closed forms contexts.
