Verified Answer :

f(x) = 2ln(x) + c [integral of 1/x is ln x] 
f(e^(1/2)) = 2ln(e^(1/2)) + c = 5 
2(1/2) + c = 5 [ln(e^a) = a] 
1 + c = 5 
c = 4 

So: 
f(e) = 2ln(e) + 4 = 2 + 4 = 6 

Answer is 6. 

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The function value of f(e) is "6".

Finding the function  f(e):

Equation:

\(\to f'(x) = \frac{2}{x} \\\\\to f(\sqrt{e}) =5\\\\\to f(e)=?\)

calculation:

\(\to \int f'(x)=\int \frac{2}{x}\ dx\\\\\to f(x)= 2 \log(x)+c.............(1)\\\\\therefore\\\\\to f(\sqrt{e})=5\\\\\to f(\sqrt{e})= 2\log(\sqrt{e}+c)\\\\5 =2 \log e^{\frac{1}{2}}+c\\\\5 =2 \times \frac{1}{2} \long e+c\\\\5=1+c\\\\c=4\\\\\)

Substition from equation (1)\\\\

\(f(x)= 2 \log (x)+4\\\\\)

now at x=e\\\\

\(f(e)= 2\log(e)+4\\\\f(e)= 2(1)+4\\\\f(e)= 6\\\\\)

Therefore, the function value of f(e) is "6".

Find out more information about the function here:

brainly.com/question/26638288

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