Showing that four stationary points of a multivariable function are at specific points.

A surface is described by the multivariable function f(x,y) where: f(x,y) = x^3 + y^3 + 9(x^2 + y^2) + 12xy a) Show that the four stationary points of this function are located at: (x1, y1) = (0, 0) (x2, y2) = (-10, -10) (x3, y3) = (-4, 2) (x4, y4) = (2, -4)

real analysis problem

We are learning Rolle, Lagrange, Fermat, Taylor Theorems in our Real Analysis class. We just finished continuity and are now studying differentiation. We are using the books by Rudin, Ross, Morrey/Protter. ****************************************************** Let f: [a,b] --> R, a < b, twice differentiable with the second derivative continuous such that f(a)=f(b)=0. Denote M = sup |f "(x)| where x is in [a,b] and g:[a,b] --> R, g(x)=(1/2)(x-a)(b-x) i) Prove that for all x in [a,b], there exists Cx in (a,b) such that f(x)= - f "(Cx)g(x). ii) Prove that if there exists x0 in (a,b) such that |f(x0)| = Mg(x0), then f = Mg or f=-Mg. **********************************... click for more

real analysis problem

Based on the Rolle, Lagrange, Fermat and Taylor Theorems. ****************************************************** Let f: [a,b] --> R, a < b, twice differentiable with the second derivative continuous such that f(a)=f(b)=0. Denote M = sup |f "(x)| where x is in [a,b] and g:[a,b] --> R, g(x)=(1/2)(x-a)(b-x) i) Prove that for all x in [a,b], there exists Cx in (a,b) such that f(x)= - f "(Cx)g(x). Cx is a constant dependent on x

Real Analysis Problem

We are learning Rolle, Lagrange, Fermat, Taylor Theorems in our Real Analysis class. We just finished continuity and are now studying differentiation. We are using the books by Rudin, Ross, Morrey/Protter. ****************************************************** Let f: [a,b] --> R, a < b, twice differentiable with the second derivative continuous such that f(a)=f(b)=0. Denote M = sup |f "(x)| where x is in [a,b] and g:[a,b] --> R, g(x)=(1/2)(x-a)(b-x) ii) Prove that if there exists x0 in (a,b) such that |f(x0)| = Mg(x0), then f = Mg or f=-Mg. **************************************************** x0 is a particular x in (a,b)

Real Analysis Problem

We have learned Rolle, Lagrange, Fermat, Taylor Theorems in our Real Analysis class and we have finished differentiation. We just started integration. In this problem we are not supposed to use any material we haven't learned, ie integration. We are using the books by Rudin, Ross, Morrey/Protter. ****************************************************** Let f:(-1,1) --> R, an odd function [f(-x) = -f(x)], five times differentiable. Prove that for all x in (-1,1), there exists theta (dependent on x) in (0,1) such that: f(x) = (1/3)(x)[f '(x)+2f '(0)]-(1/180)(x^5)(f'''''(theta(dependent on x)x)) ****************************************************** - theta(dependent on x) is symbo... click for more