Given the equation: $3f(x) + 2f\left(\frac{1}{x}\right) = \frac{1}{x} - 10$ Replace $x$ with $\frac{1}{x}$ in the original equation: $3f\left(\frac{1}{x}\right) + 2f(x) = x - 10$ Now, we have two equations: $3f(x) + 2f\left(\frac{1}{x}\right) = \frac{1}{x} - 10$ $3f\left(\frac{1}{x}\right) + 2f(x) = x - 10$ By adding the two equations, we can find $f(x)$: $5f(x) = \frac{3}{x} - 2x - 10$ Now, let's differentiate both sides with respect to $x$: $5f'(x) = -\frac{3}{x^2} - 2$ Now, we can find the values for $f(3)$ and $f'\left(\frac{1}{4}\right)$: $f(3) = \frac{1}{5}(1 - 6 - 10) = -3$ $f'\left(\frac{1}{4}\right) = \frac{1}{5}(-48 - 2) = -10$ Finally, calculate the expression we are interested in : $\left|f(3) + f'\left(\frac{1}{4}\right)\right| = \left|-3 - 10\right| = 13$