MATH224: Studio 6 Assignment (Second Derivatives Test)
Created by: Gus Greivel , Last Modified: 10/1/2015, 9:50pm
Background:
We discussed the second derivatives test (Wikepedia – Second Derivatives Test) in class this
week. This test allowed us to classify critical points of a twice differentiable function of two
variable as (i) local minima, (ii) local maxima or (iii) saddle points (Wikepedia – Maxima and
Minima) using the second partials of the function. Using the result derived in class this week
please do the following for the functions given below:
Practice Problems: [The following technique is demonstrated in this video.]
1. Identify the critical points of the function by finding and setting the first partials equal to
zero.
2. Find the second partial of the function and construct the determinant of the Hessian
(Wikepedia – Hessian Matrix)…Not the same as (Wikepedia – Hessian Soldier)
D=
f xx
f xy
f yx
f yy
3. Evaluate D and fxx at each critical point and classify that critical point as either (i) a local
minimum, (ii) a local maximum, or (iii) a saddle point).
4. Validate your hand-calculated work and plot the surface to illustrate these points using the
Mathematica notebook for this week’s studio.
Functions:
(1)
f (x, y) = 4 - xy -
x 4 y4
4 4
(2) h(x, y) = x 2 y2 - 2xy2 + 3x 2 y - 6xy = xy(x - 2)(y + 3)
(3) g(x, y) = 2xye(-x -y
2
2
)
Reflection Question:
Considering the values of the individual second partials (fxx, fyy, fxy, fyx) and of D at each of the
critical points for the functions above and how those critical points are characterized (min, max,
saddle), give an explanation of the types of relationships between the second partials that might
lead to classifying a critical point as a local maximum, a local minimum or a saddle point.
Challenge Problem:
Define your own twice differentiable function – with at least three critical points – and replicate
the Mathematica validation (step 4 above) to identify and classify the critical points of your
function.
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