Answer:
f(x) = (2/3)xln(x) - (2/3)x + 1
steps
To find an expression for f(x), we need to find the antiderivative of f'(x) = (2/3)ln(x)/x.
By using the antiderivative rule, we get
f(x) = (2/3)xln(x) - (2/3)x + C.
Now we can use the point (1,1) as an initial condition to find the value of C.
We know that f(1) = 1, so we can substitute this into the equation we found for f(x) above:
1 = (2/3) * 1 * ln(1) - (2/3) * 1 + C
By solving this equation, we find that C = 1.
Therefore, the expression for f(x) is:
f(x) = (2/3)xln(x) - (2/3)x + 1
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