The set of points where f(x) = frac{4x}{5 + 6 | vedclass

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(C) Given $f(x) = \frac{4x}{5 + 6|x|}$.We can define $f(x)$ as:$f(x) = \begin{cases} \frac{4x}{5 - 6x}, & x < 0 \ \frac{4x}{5 + 6x}, & x \geq 0 \end{

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$( - \infty, \infty)$

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(C) Given $f(x) = \frac{4x}{5 + 6|x|}$.We can define $f(x)$ as:$f(x) = \begin{cases} \frac{4x}{5 - 6x}, & x To check differentiability at $x = 0$,we find the left-hand derivative $(LHD)$ and right-hand derivative $(RHD)$:$f'(x) = \begin{cases} \frac{4(5 - 6x) - 4x(-6)}{(5 - 6x)^2} = \frac{20}{(5 - 6x)^2}, & x  0 \end{cases}$Now,$f'(0^-) = \lim_{x 	o 0^-} \frac{20}{(5 - 6x)^2} = \frac{20}{25} = \frac{4}{5}$.And $f'(0^+) = \lim_{x 	o 0^+} \frac{20}{(5 + 6x)^2} = \frac{20}{25} = \frac{4}{5}$.Since $f'(0^-) = f'(0^+) = \frac{4}{5}$,the function is differentiable at $x = 0$.Since the function is a rational function with a non-zero denominator for all $x \in \mathbb{R}$,it is differentiable everywhere.Thus,the set of points is $( - \infty, \infty)$.

The set of points where $f(x) = \frac{4x}{5 + 6|x|}$ is differentiable is:

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