Homogeneous Function Degree 4 Solution
This problem involves Euler's Homogeneous Function Theorem, which relates a homogeneous function to its partial derivatives.
Euler's Theorem Statement
If $f(x, y)$ is a continuously differentiable homogeneous function of degree $n$, then it satisfies the following identity:
$x \frac{\partial f(x, y)}{\partial x} + y \frac{\partial f(x, y)}{\partial y} = n f(x, y)$
Applying to the Problem
The question states that $f(x, y)$ is a homogeneous function of degree 4. Thus, we have $n = 4$. Substituting $n=4$ into Euler's theorem gives:
$x \frac{\partial f(x, y)}{\partial x} + y \frac{\partial f(x, y)}{\partial y} = 4f(x, y)$
Result
This identity must necessarily be true for the given function. Comparing this result with the options provided, it matches Option 3.
Important Questions from Algebra
- If $Pe^x = Qe^{-x}$ for all real values of $x$, which one of the following statements is true?
The relationship between two variables $x$ and $y$ is given by $x + py + q = 0$ and is shown in the figure. Find the values of $p$ and $q$.
Note: The figure shown is representative.
The real variables $x, y, z$ and the real constants $p, q, r $ satisfy
$\frac{x}{pq - r^2} = \frac{y}{qr - p^2} = \frac{z}{rp - q^2}$
Given the denominators are non-zero, the value of $px + qy + rz$ isThe complex function
$e^{-\left(\frac{2}{z-1}\right)}$
has __________________Consider two matrices: $P = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ and $Q = \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}$.
Which of the following statement is/are true?
