Animated Solution for Mathematics - Differentiation: For the differentiable function $f:\mathbb{R}-\{0\}\to \mathbb{R}$, let $3f(x)+2f\left(\frac{1}{x}\right)=\frac{1}{x}-10$, then $|f(3)+f^{\prime }(1/4)|$ is equal to

Imagine you are standing before a mirror, looking at your own reflection. In the world of mathematics, functional equations often present us with a similar scenario. When you see a problem involving $f(x)$ and $f(\frac{1}{x})$, you are essentially looking at a mathematical mirror.

The function is reflecting itself across the transformation $x\to \frac{1}{x}$. This is not just a coincidence; it is a structural hint. Our journey begins with the given equation:

By replacing every instance of $x$ with $\frac{1}{x}$, we generate a second, perfectly complementary equation:

Now, we have a system. Think of $f(x)$ and $f(\frac{1}{x})$ as two unknown variables. This is the moment where the complexity collapses into simplicity.

We want to isolate $f(x)$. To do this, we need to eliminate $f(\frac{1}{x})$. We multiply our first equation by $3$ and our second equation by $2$:

When we subtract the second equation from the first, the terms involving $f(\frac{1}{x})$ vanish entirely. We are left with:

We have successfully isolated our function. It is no longer a mystery; it is an explicit expression:

Now that we have the explicit form of $f(x)$, we are ready to tackle the second part of our challenge: finding $f^{\prime }(\frac{1}{4})$. We differentiate our expression with respect to $x$:

With this derivative in hand, we substitute $x=\frac{1}{4}$:

This yields the result $f^{\prime }(\frac{1}{4})=-10$.

Finally, we return to our original goal: $|f(3)+f^{\prime }(\frac{1}{4})|$. First, we find $f(3)$ by substituting $x=3$ into our isolated function:

Adding these together, we get:

The elegance of this solution lies in how the symmetry allowed us to break down a complex functional relationship into a simple algebraic system, and then into a straightforward calculus problem.