2. Suppose that f(x,y) is a continuous function of two variables such that each of the first and second order partial derivatives are also continuous at the point (x0, Yo). (a) Consider the polynomial Q(x, y) = f(xo; Yo) + f«(xo, Yo)(x – co)+ fy(xo,Yo)(y – yo) 1 1 +;fe (To, Yo)(x – xo)² + fæy(T0, Y0)(x – xo)(y – yo) + ¿ fyy(y – yo)² 2 Show that f and Q have the same values, same first derivatives, and same second derivatives at the point (xo, Yo). (b) Briefly explain – maybe using the Calculus II idea of a Taylor polynomial? - why we might expect that f(x, y) × Q(x, y) for (x, y) × (xo, Yo). (c) Let g(x, y) = f(x+ x0,Y+ Yo) – f(xo, Yo). and suppose that (x0, Yo) is a critical point of f. Show: i. g(0,0) = 0 ii. ga (0,0) = 0 = 9y(0, 0). iii. Show that the second partials of g, evaluated at (0,0) give the same answer as the second partials of f, evaluated at (x0, yo). (d) Explain why the behavior of f at (xo, Yo) is the same as the behavior of g at (0,0) – that is, both have a local minimum, both have a local maximum, or both have a saddle point. [Hint: g is really just a shift of f.) (e) Build the second-order polynomial approximation for g at (0, 0). Use the result of question #1 to explain why we can apply the Second Derivative Test to this polynomial, and hence to g, to classify the critical point at (0,0). (f) Finally, summarize the result of this problem, as it relates to classifying the critical point (xo, Yo) for f.
2. Suppose that f(x,y) is a continuous function of two variables such that each of the first and second order partial derivatives are also continuous at the point (x0, Yo). (a) Consider the polynomial Q(x, y) = f(xo; Yo) + f«(xo, Yo)(x – co)+ fy(xo,Yo)(y – yo) 1 1 +;fe (To, Yo)(x – xo)² + fæy(T0, Y0)(x – xo)(y – yo) + ¿ fyy(y – yo)² 2 Show that f and Q have the same values, same first derivatives, and same second derivatives at the point (xo, Yo). (b) Briefly explain – maybe using the Calculus II idea of a Taylor polynomial? - why we might expect that f(x, y) × Q(x, y) for (x, y) × (xo, Yo). (c) Let g(x, y) = f(x+ x0,Y+ Yo) – f(xo, Yo). and suppose that (x0, Yo) is a critical point of f. Show: i. g(0,0) = 0 ii. ga (0,0) = 0 = 9y(0, 0). iii. Show that the second partials of g, evaluated at (0,0) give the same answer as the second partials of f, evaluated at (x0, yo). (d) Explain why the behavior of f at (xo, Yo) is the same as the behavior of g at (0,0) – that is, both have a local minimum, both have a local maximum, or both have a saddle point. [Hint: g is really just a shift of f.) (e) Build the second-order polynomial approximation for g at (0, 0). Use the result of question #1 to explain why we can apply the Second Derivative Test to this polynomial, and hence to g, to classify the critical point at (0,0). (f) Finally, summarize the result of this problem, as it relates to classifying the critical point (xo, Yo) for f.
Linear Algebra: A Modern Introduction
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ISBN:9781285463247
Author:David Poole
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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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![2. Suppose that f(x, y) is a continuous function of two variables such that each of the first and second order
partial derivatives are also continuous at the point (x0, Yo).
(a) Consider the polynomial
Q(x, y) = f(xo, Yo) + fæ(xo,Yo)(x – xo) + fy(xo, Yo)(y – yo)
1
1
+faz (to, Yo)(x – xo)² + fay(xo, Yo)(x – xo)(y – yo) + „fvv (y – yo)²
Show that f and Q have the same values, same first derivatives, and same second derivatives at the point
(xo, Yo).
(b) Briefly explain – maybe using the Calculus II idea of a Taylor polynomial? – why we might expect that
f (x, y) - Q(x, y) for (x, y) × (x0, yo).
(c) Let g(x, y) = f(x+ x0,y+ Yo) – f(xo, Yo). and suppose that (xo, Yo) is a critical point of f. Show:
i. g(0,0) = 0
ii. ga (0, 0) = 0 = 9y(0, 0).
iii. Show that the second partials of g, evaluated at (0,0) give the same answer as the second partials of
f, evaluated at (xo, Yo).
(d) Explain why the behavior of f at (xo, yo) is the same as the behavior of
local minimum, both have a local maximum, or both have a saddle point. [Hint: g is really just a shift of
f.]
g at (0,0) – that is, both have a
(e) Build the second-order polynomial approximation for g at (0,0). Use the result of question #1 to explain
why we can apply the Second Derivative Test to this polynomial, and hence to g, to classify the critical
point at (0,0).
(f) Finally, summarize the result of this problem, as it relates to classifying the critical point (xo, Yo) for f.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1863dd2d-66ae-4de0-8154-3d7d6de9d0c9%2F6a8ad1b4-a4c6-451b-b7b8-38dedf0faf3a%2Fwo9hpo_processed.png&w=3840&q=75)
Transcribed Image Text:2. Suppose that f(x, y) is a continuous function of two variables such that each of the first and second order
partial derivatives are also continuous at the point (x0, Yo).
(a) Consider the polynomial
Q(x, y) = f(xo, Yo) + fæ(xo,Yo)(x – xo) + fy(xo, Yo)(y – yo)
1
1
+faz (to, Yo)(x – xo)² + fay(xo, Yo)(x – xo)(y – yo) + „fvv (y – yo)²
Show that f and Q have the same values, same first derivatives, and same second derivatives at the point
(xo, Yo).
(b) Briefly explain – maybe using the Calculus II idea of a Taylor polynomial? – why we might expect that
f (x, y) - Q(x, y) for (x, y) × (x0, yo).
(c) Let g(x, y) = f(x+ x0,y+ Yo) – f(xo, Yo). and suppose that (xo, Yo) is a critical point of f. Show:
i. g(0,0) = 0
ii. ga (0, 0) = 0 = 9y(0, 0).
iii. Show that the second partials of g, evaluated at (0,0) give the same answer as the second partials of
f, evaluated at (xo, Yo).
(d) Explain why the behavior of f at (xo, yo) is the same as the behavior of
local minimum, both have a local maximum, or both have a saddle point. [Hint: g is really just a shift of
f.]
g at (0,0) – that is, both have a
(e) Build the second-order polynomial approximation for g at (0,0). Use the result of question #1 to explain
why we can apply the Second Derivative Test to this polynomial, and hence to g, to classify the critical
point at (0,0).
(f) Finally, summarize the result of this problem, as it relates to classifying the critical point (xo, Yo) for f.
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