To find the value of [tex]\(\lceil f(4) \rceil\)[/tex] where [tex]\(f(x) = \frac{1}{3} \cdot 4^x\)[/tex]:
First, we need to evaluate the function [tex]\(f(x)\)[/tex] at [tex]\(x = 4\)[/tex]:
[tex]\[ f(4) = \frac{1}{3} \cdot 4^4 \][/tex]
Calculate the expression inside the function step by step:
1. Calculate [tex]\(4^4\)[/tex]:
[tex]\[ 4^4 = 4 \cdot 4 \cdot 4 \cdot 4 = 256 \][/tex]
2. Divide the result by 3:
[tex]\[ \frac{256}{3} \approx 85.33333333333333 \][/tex]
So, we have:
[tex]\[ f(4) = 85.33333333333333 \][/tex]
Next, we need to find the ceiling of this value.
The ceiling function, denoted by [tex]\(\lceil x \rceil\)[/tex], gives the smallest integer that is greater than or equal to [tex]\(x\)[/tex]. Therefore,
[tex]\[ \lceil 85.33333333333333 \rceil = 86 \][/tex]
Thus, the value of [tex]\(\lceil f(4) \rceil\)[/tex] is [tex]\(86\)[/tex].
