IGNOU MTE 7 2026 - State a necessary condition for the functional dependence of two diffe

Question

State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of $\mathbb{R}^2$ . Verify this theorem for the functions f and g, defined by

$$f(x, y) = \frac{y - x}{y + x}, \quad g(x, y) = \frac{x}{y}.$
(b) Using the Implicit Function Theorem, show that there exists a unique differentiable function g in a neighbourhood of 1 such that $g(1) = 2$ and $F(g(y), y) = 0$ in a neighbourhood of (2, 1), where

$$F(x, y) = x^5 + y^5 - 16xy^3 - 1 = 0$

defines the function F. Also find g'(y).
(c) Check the local invertibility of the function f defined by $f(x, y) = (x^2 - y^2, 2xy)$ at (1, -1). Find a domain for the function f in which f is invertible.

09 Jan 2026

Answer :

Word Count : 228

Two differentiable functions (f) and (g) on an open set (D \subset \mathbb{R}^2) are said to be functionally dependent if there exists a differentiable function (\Phi) such that (\Phi(f(x,y),g(x,y))=0) for all ((x,y)\in D). A necessary condition for functional dependence is that the Jacobian determinant [ J(f,g)=\frac{\partial(f,g)}{\partial(x,y)}=f_x g_y - f_y g_x ] vanishes identically on (D).

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