Question

given \\(f(x)\\) to be a differentiable function such that

\\f(x) = \frac{x}{x + 4},\\

find the values of \\(x\\) where \\(f(x) = 3\\). list each exact answer separated by a comma, decimal approximations are not accepted.

\\(x =\\)

Explanation:

Differentiate the function using the quotient rule

$$ f'(x) = \frac{\frac{d}{dx}[x] \cdot (x+4) - x \cdot \frac{d}{dx}[x+4]}{(x+4)^2} = \frac{1 \cdot (x+4) - x \cdot 1}{(x+4)^2} = \frac{4}{(x+4)^2} $$

Set the derivative equal to 3 and solve for x

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Format the final exact values

$$ x = -4 + \frac{2\sqrt{3}}{3}, -4 - \frac{2\sqrt{3}}{3} $$

Answer:

Given \(f(x)\) to be a differentiable function such that

$$f(x) = \frac{x}{x+4},$$

find the values of \(x\) where \(f'(x) = 3\). List each exact answer separated by a comma, decimal approximations are not accepted.

\(x =\) <blank>\(-4 + \frac{2\sqrt{3}}{3}, -4 - \frac{2\sqrt{3}}{3}\)</blank>