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larapoghosyan91
17.08.2021 •
Mathematics
Let the (x; y) coordinates represent locations on the ground. The height h of
a particle is governed by the function described below. Classify all the critical
points of height and find the value(s) of height h at those points.
h(x; y) = 8x + 10y - 4xy - 4x^2 - 4y^3
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Ответ:
The critical points of h(x,y) occur wherever its partial derivatives
and
vanish simultaneously. We have
Substitute y in the second equation and solve for x, then for y :
This is to say there are two critical points,
To classify these critical points, we carry out the second partial derivative test. h(x,y) has Hessian
whose determinant is
. Now,
• if the Hessian determinant is negative at a given critical point, then you have a saddle point
• if both the determinant and
are positive at the point, then it's a local minimum
• if the determinant is positive and
is negative, then it's a local maximum
• otherwise the test fails
We have
while
So, we end up with
Ответ:
-1,-1,1
Step-by-step explanation: