The value of f'(x) is [tex]f'(1) = 1.303[/tex], and the estimate should be less than f'(x)
How to determine the estimate
The equation of the function is given as:
[tex]f(x) = 3\log(x)[/tex]
To estimate f'(1), we set x = 1 and a number close to 1 (for instance x = 1.0001).
Next, we compute the average rate of change of this interval.
So, we have:
[tex]f'(1) = \frac{f(1.0001) - f(1)}{1.0001 -1}[/tex]
[tex]f'(1) = \frac{f(1.0001) - f(1)}{0.0001}[/tex]
Calculate f(1.0001) and f(1)
[tex]f(1.0001) = 3*log(1.0001) =0.00013028183[/tex]
[tex]f(1.0001) = 3*log(1) =0[/tex]
So, we have:
[tex]f'(1) = \frac{0.00013028183 - 0}{0.0001}[/tex]
[tex]f'(1) = \frac{0.00013028183}{0.0001}[/tex]
[tex]f'(1) = 1.3028183[/tex]
Approximate
[tex]f'(1) = 1.303[/tex]
From the graph of f(x) (see attachment), we have that:
The graph of function f(x) is concave down
This means that, the estimate should be less than f'(x)
Read more about logarithmic functions at:
https://brainly.com/question/13473114
