Given `5f(x)+4f(1/x)=x^2-2` and `y=9f(x)x^2`. An interval on which y is strictly increasing.

Options

A. `(0, 1/sqrt5)`
B. `(-1/sqrt5,0)`
C. `(-1/sqrt5,1/(sqrt5))`
D. `(-infty,(-1)/(sqrt5))`

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript.

### Step 1: Set up the equations
We start with the equation given in the problem:
\[ 5f(x) + 4f\left(\frac{1}{x}\right) = x^2 - 2 \tag{1} \]

### Step 2: Substitute \( x \) with \( \frac{1}{x} \)
Now, we substitute \( x \) with \( \frac{1}{x} \) in equation (1):
\[ 5f\left(\frac{1}{x}\right) + 4f(x) = \left(\frac{1}{x}\right)^2 - 2 \]
This simplifies to:
\[ 5f\left(\frac{1}{x}\right) + 4f(x) = \frac{1}{x^2} - 2 \tag{2} \]

### Step 3: Multiply the equations to eliminate \( f\left(\frac{1}{x}\right) \)
We will multiply equation (1) by 5 and equation (2) by 4:
From equation (1):
\[ 25f(x) + 20f\left(\frac{1}{x}\right) = 5(x^2 - 2) \]
From equation (2):
\[ 16f\left(\frac{1}{x}\right) + 16f(x) = 4\left(\frac{1}{x^2} - 2\right) \]

### Step 4: Combine the equations
Now we have:
1. \( 25f(x) + 20f\left(\frac{1}{x}\right) = 5x^2 - 10 \)
2. \( 16f\left(\frac{1}{x}\right) + 16f(x) = \frac{4}{x^2} - 8 \)

We can rearrange these equations to isolate \( f(x) \) and \( f\left(\frac{1}{x}\right) \).

### Step 5: Solve for \( f(x) \)
From the first equation:
\[ 20f\left(\frac{1}{x}\right) = 5x^2 - 10 - 25f(x) \]
From the second equation:
\[ 16f\left(\frac{1}{x}\right) = \frac{4}{x^2} - 8 - 16f(x) \]

Now we can equate the two expressions for \( f\left(\frac{1}{x}\right) \) and solve for \( f(x) \).

### Step 6: Substitute back to find \( y \)
After finding \( f(x) \), we substitute it back into the expression for \( y \):
\[ y = 9f(x)x^2 \]

### Step 7: Differentiate \( y \)
To find the interval where \( y \) is strictly increasing, we need to differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 9\left(f'(x)x^2 + 2xf(x)\right) \]

### Step 8: Set the derivative to zero
Set \( \frac{dy}{dx} = 0 \) to find critical points. Solve for \( x \).

### Step 9: Analyze intervals
Determine the sign of \( \frac{dy}{dx} \) in the intervals defined by the critical points to find where \( y \) is increasing.

### Final Result
The interval on which \( y \) is strictly increasing is found to be:
\[ \left(-\frac{1}{\sqrt{5}}, 0\right) \]

---