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
- For positive non-zero real variables $x$ and $y$, if
$ln\left(\frac{x+y}{2}\right) = \frac{1}{2} [ln\left(x\right) + ln\left(y\right)]$
then, the value of $\frac{x}{y} + \frac{y}{x}$ is Given $f(x, y) = x^2 - 2xy + y^2$
The complete contour of the equation $f(x, y) = 1$ is described by the option(s) ___.
- It is given that $x$ and $y$ are integers in the following equation:
$$(x + y - 7)^2 + (y + 3x - 13)^2 = 0$$
The value of $(x^3 + y^3)$ is ________ (in integer). - If $pqr \neq 0$ and $p^{-x} = \frac{1}{q}$, $q^{-y} = \frac{1}{r}$, $r^{-z} = \frac{1}{p}$, what is the value of the product $xyz$?
- Two points $(4, p)$ and $(0, q)$ lie on a straight line having a slope of $3/4$. The value of $(p – q)$ is
