For the differentiable function \[f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}\], let \[3 f(x)+2 f\left(\frac{1}{x}\right)=\frac{1}{x}-10\], then \[\left|f(3)+f^{\prime}\left(\frac{1}{4}\right)\right|\] is equal to
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Official solution
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 :
\[\left|f(3) + f'\left(\frac{1}{4}\right)\right| = \left|-3 - 10\right| = 13\]