83. A non-zero function f (x) is symmetrical about the line `y=x` then the value of `lambda` (constant) such that `f^2(x)=(f^-1(x))^2-lambdaxf(x)f^-1(x)+3x^2f(x)` where all x`in R^+`

Answer

To solve the problem, we need to find the value of the constant \( \lambda \) such that the equation

\[ f^2(x) = (f^{-1}(x))^2 - \lambda x f(x) f^{-1}(x) + 3x^2 f(x) \]

holds true, given that the function \( f(x) \) is symmetrical about the line \( y = x \).

### Step-by-Step Solution:

1. **Understanding Symmetry**:
Since \( f(x) \) is symmetrical about the line \( y = x \), we have:
\[ f(x) = f^{-1}(x) \]
This means that the function and its inverse are the same.

2. **Substituting the Inverse**:
We can substitute \( f^{-1}(x) \) with \( f(x) \) in the given equation:
\[ f^2(x) = (f(x))^2 - \lambda x f(x) f(x) + 3x^2 f(x) \]

3. **Simplifying the Equation**:
The equation simplifies to:
\[ f^2(x) = f^2(x) - \lambda x f^2(x) + 3x^2 f(x) \]

4. **Rearranging Terms**:
Move all terms involving \( f^2(x) \) to one side:
\[ f^2(x) - f^2(x) + \lambda x f^2(x) = 3x^2 f(x) \]
This simplifies to:
\[ \lambda x f^2(x) = 3x^2 f(x) \]

5. **Dividing by \( f(x) \)**:
Since \( f(x) \) is a non-zero function, we can divide both sides by \( f(x) \):
\[ \lambda x f(x) = 3x^2 \]

6. **Isolating \( \lambda \)**:
Now, we can isolate \( \lambda \):
\[ \lambda = \frac{3x^2}{x f(x)} = \frac{3x}{f(x)} \]

7. **Finding \( \lambda \)**:
To find a specific value for \( \lambda \), we can assume a simple case where \( f(x) = x \). If \( f(x) = x \), then:
\[ \lambda = \frac{3x}{x} = 3 \]

### Conclusion:
Thus, the value of \( \lambda \) is:
\[ \boxed{3} \]