Answered: THEOREM 15.14 Derivatives and Local…

THEOREM 15.14 Derivatives and Local Maximum/Minimum Values If f has a local maximum or minimum value at (a, b) and the partial derivatives f, and f, exist at (a, b), then f_(a, b) = f,(a, b) = 0.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter9: Multivariable Calculus

Section9.3: Maxima And Minima

Problem 2YT

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Question

The paraboloid z = x² + y² - 4x + 2y + 5 has a local minimum at (2, -1). Verify the conclusion of as shown for this function.

THEOREM 15.14 Derivatives and Local Maximum/Minimum Values
If f has a local maximum or minimum value at (a, b) and the partial derivatives f,
and f, exist at (a, b), then f_(a, b) = f,(a, b) = 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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