(a) Consider the function f: R² → R where f(x, y) = cos(x²y). (i) Find all the first and second order partial derivatives. (You may assume that fry = fyr.) (ii) Calculate the differential df of the function f(x, y) at the points (a,0), a € R and (0, b), b € R. (iii) Compute the directional derivative of f at (1, π/2) in the direction of v = i + j. (b) Consider the function g(x, y) = x² + 2xy + 2xy². (i) Show that (0, 0) is a critical point and find any other critical point(s) of g. (ii) Classify the critical point (0,0) of g(x, y) as a local maximum a local minimum or a

(a) Consider the function f: R² → R where f(x, y) = cos(x²y). (i) Find all the first and second order partial derivatives. (You may assume that fry = fyr.) (ii) Calculate the differential df of the function f(x, y) at the points (a,0), a € R and (0, b), b € R. (iii) Compute the directional derivative of f at (1, π/2) in the direction of v = i + j. (b) Consider the function g(x, y) = x² + 2xy + 2xy². (i) Show that (0, 0) is a critical point and find any other critical point(s) of g. (ii) Classify the critical point (0,0) of g(x, y) as a local maximum a local minimum or a

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...

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