IGNOU MTE 7 2026 - ) Does the function satisfy the requirement of Schwarz&#3

Question

) Does the function

satisfy the requirement of Schwarz's theorem at (1, 1)? Justify your answer.
$\quad$ (b) Locate and classify the stationary points of the following:
$\quad$ (i) $f(x, y) = 4xy + x^4 - y^4$
$\quad$ (ii) $f(x, y) = xy + \frac{2}{x} + \frac{4}{y}, x > 0, y > 0$

09 Jan 2026

Answer :

Word Count : 385

For part (a), we check if the function satisfies Schwarz's theorem (equality of mixed partial derivatives) at ((1,1)). Let the function be (f(x,y)). Compute the second-order partial derivatives: [ f_{xy} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right), \quad f_{yx} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) ] Evaluate both at ((1,1)). If (f_{xy}(1,1) = f_{yx}(1,1)), the theorem is satisfied. If not, it fails. Without the explicit function given in your query, the method is: compute both mixed partials and compare their values at (1,1). --- For part (b)(i), (f(x,y) = 4xy + x^4 - y^4). 1. Compute first-order partials: [ f_x = 4y + 4x^3, \quad f_y = 4x - 4y^3 ] 2. Set them equal to zero for stationary points: [ 4y + 4x^3 = 0 \implies y = -x^3

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