# Using the approximation, explain why the second derivative test works.

by jumboopizza
Tags: approximation, derivative, explain, test
 P: 13 [b]1. The problem statement, all variables and given/known data[/ Using the approximation, explain why the second derivative test works approximation=f(x0+delta x, y0+delta y) delta x and delta y are small... 2. Relevant equations f(x0+delta x,y0+delta y) 3. The attempt at a solution ok so i know the first derivative of it is: fx(x0+y0)*delta x+fy(x0+y0)*delta y and second derivative is: fxx(x0,y0)*delta x^2+fxy(x0,y0)*delta x*delta y+fyy(x0,y0)*delta y^2 it justs seems like since delta x and delta y are getting smaller,that everything will be going towards 0....so why does it work? 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution
 Emeritus Sci Advisor HW Helper PF Gold P: 7,076 What is the problem as written? f appears to be a function of two variables. What is the second derivative test for a function of two variables?
 P: 13 Using the approximation, explain why the second derivative works. Give three exam- ples for each scenario of the second derivative test. isnt that what the approximation is? the f(x+delta x,y +delta y)? its asking about finding local mins,local max and saddle points....now i can show those examples,but how can i explain how it works?its like a proof or something
HW Helper
P: 4,152

## Using the approximation, explain why the second derivative test works.

 Quote by jumboopizza [b]1. The problem statement, all variables and given/known data[/ Using the approximation, explain why the second derivative test works approximation=f(x0+delta x, y0+delta y) delta x and delta y are small... 2. Relevant equations f(x0+delta x,y0+delta y) 3. The attempt at a solution ok so i know the first derivative of it is: fx(x0+y0)*delta x+fy(x0+y0)*delta y and second derivative is: fxx(x0,y0)*delta x^2+fxy(x0,y0)*delta x*delta y+fyy(x0,y0)*delta y^2 it justs seems like since delta x and delta y are getting smaller,that everything will be going towards 0....so why does it work? 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution
You don't say what you are testing for. If you mean the second-order test for a maximum or minimum, you still need to be more specific: the necessary conditions and the (most common) sufficient conditions are a bit different. You need to tell us which ones you want.

RGV
 Emeritus Sci Advisor HW Helper PF Gold P: 7,076 Mathematical language is very exacting. You need to say what you mean & mean what you say. f(x0+Δx, y0+Δy) is the (exact) value of the function, f, at the point (x0+Δx, y0+Δy). If the first derivatives of f are zero at the point, (x0, y0), then the following is an approximation to f at the point (x0+Δx, y0+Δy). f(x0+Δx, y0+Δy) ≈ f(x0, y0) + (1/2)[ fxx(x0, y0)(Δx)2 +2 fxy(x0, y0)(Δx)(Δy) + fyy(x0, y0)(Δy)2 ]

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