Find the interval in which the function f(x) | vedclass

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(D) Given $f(x) = \log x - \frac{2x}{x+2}$.The domain of the function is $x > 0$.Differentiating with respect to $x$:$f'(x) = \frac{1}{x} - \frac{(x+2

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$x \in(0, \infty)$

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(D) Given $f(x) = \log x - \frac{2x}{x+2}$.The domain of the function is $x > 0$.Differentiating with respect to $x$:$f'(x) = \frac{1}{x} - \frac{(x+2)(2) - 2x(1)}{(x+2)^2}$$f'(x) = \frac{1}{x} - \frac{4}{(x+2)^2}$$f'(x) = \frac{(x+2)^2 - 4x}{x(x+2)^2} = \frac{x^2 + 4x + 4 - 4x}{x(x+2)^2} = \frac{x^2 + 4}{x(x+2)^2}$Since $x^2 + 4 > 0$ and $(x+2)^2 > 0$ for all $x$ in the domain,$f'(x) > 0$ for all $x > 0$.Therefore,the function is strictly increasing for $x \in (0, \infty)$.

Find the interval in which the function $f(x) = \log x - \frac{2x}{x+2}$ is strictly increasing.

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