Engineering Calculus Notes 370

Engineering Calculus Notes 370 - f : R 3 R has a local...

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358 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION (b) How many distinct partial derivatives of order n can a C n function of two variables have? (c) How many distinct partial derivatives of order n can a C n function of three variables have? 7. Prove the comment in the proof of Proposition 3.7.2 , that max ( i,j ) |△ x i x j | b△ −→ x b 2 1 . Challenge problem: 8. Consider the function of two variables f ( x,y ) = b xy ( x 2 y 2 ) x 2 + y 2 if x 2 + y 2 n = 0 , 0 at ( x,y ) = (0 , 0) . (a) Calculate ∂f ∂x ( x,y ) and ∂f ∂y ( x,y ) for ( x,y ) n = (0 , 0). (b) Calculate ∂f ∂x (0 , 0) and ∂f ∂y (0 , 0). (c) Calculate the second-order partial derivatives of f ( x,y ) at ( x,y ) n = (0 , 0). (d) Calculate the second-order partial derivatives of f ( x,y ) at the origin. Note that 2 f ∂x∂y (0 , 0) n = 2 f ∂y∂x (0 , 0) . Explain. 3.8 Local Extrema The Critical Point Theorem (Theorem 3.6.10 ) tells us that a local extremum must be a critical point: if a diFerentiable function
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Unformatted text preview: f : R 3 R has a local maximum (or local minimum) at a , then d a f ( v ) = 0 for all v R 3 . The converse is not true: for example, the function f ( x ) = x 3 has a critical point at the origin but is strictly increasing on the whole real line. Other phenomena are possible for multivariate functions: for example, the restriction of f ( xy ) = x 2 y 2 to the x-axis has a minimum at the origin, while its restriction to the y-axis has a maximum there. So to determine whether a critical point a is a local extremum, we need to study the (local) behavior in all directionsin particular, we need to study the Hessian d 2 a f ....
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