Animated Solution for Mathematics - Differentiation: If $5f(x)+4f\left(\frac{1}{x}\right)=x^2-2,\forall x\ne 0$ and $y=9x^{2}f(x)$, then $y$ is strictly increasing in :

The functional equation provided is:

To solve for $f(x)$, we recognize the symmetry between $x$ and $\frac{1}{x}$. By replacing $x$ with $\frac{1}{x}$ in the original equation, we generate a second equation:

We now have a system of two linear equations with two variables, $f(x)$ and $f\left(\frac{1}{x}\right)$. To isolate $f(x)$, we multiply the first equation by $5$ and the second equation by $4$:

Subtracting the second resulting equation from the first yields:

Substituting this into the expression $y=9x^{2}f(x)$, we obtain:

To determine where the function $y$ is strictly increasing, we must satisfy the condition $\frac{dy}{dx}>0$. Differentiating the polynomial $y=5x^4-2x^2-4$ with respect to $x$ gives:

We set this derivative to be strictly positive:

Factoring the expression further, we get:

We identify the critical points on the number line: $x=0$, $x=\frac{1}{\sqrt{5}}$, and $x=-\frac{1}{\sqrt{5}}$.

Applying the Wavy Curve method, we test the intervals. Starting from the rightmost interval $(\frac{1}{\sqrt{5}},\infty )$ with a positive sign and alternating signs across each critical point, we find the regions where the derivative is positive.

The function $y$ is strictly increasing on the intervals: