The origin is a critical point for the function f(x, y) = 15 − x^2y^2 and ∆(x, y) = 0 there, i.e., the Second Derivative Test fails. Use what you know about shapes of functions to decide if there is a local minimum, local maximum, or saddle point for this function at (0, 0).

The origin is a critical point for the function f(x, y) = 15 − x^2y^2 and ∆(x, y) = 0 there, i.e., the Second Derivative Test fails. Use what you know about shapes of functions to decide if there is a local minimum, local maximum, or saddle point for this function at (0, 0).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

The origin is a critical point for the function f(x, y) = 15 − x^2y^2 and ∆(x, y) = 0 there, i.e.,
the Second Derivative Test fails. Use what you know about shapes of functions to decide if
there is a local minimum, local maximum, or saddle point for this function at (0, 0).

Formula Formula A function f ( x ) is also said to have attained a local minimum at x = a , if there exists a neighborhood ( a − δ , a + δ ) of a such that, f ( x ) > f ( a ) , ∀ x ∈ ( a − δ , a + δ ) , x ≠ a f ( x ) − f ( a ) > 0 , ∀ x ∈ ( a − δ , a + δ ) , x ≠ a In such a case f ( a ) is called the local minimum value of f ( x ) at x = a .

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