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

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

typically if you want , you would actually put questions here for us to you. but hey, that's just me. maybe someone will magically answer the questions for you without knowing what they are.

d. 24+-0.114

step-by-step explanation:

the question requires us to calculate the approximate 95% confidence interval for the mean number of ounces of ketchup per bottle in the sample given the following statistics;

sample mean = 24

standard deviation, σ = 0.4

sample size , n = 49

the confidence interval for a population mean is calculated using the formula;

sample mean ± z-score*(σ/sqrt(n))

the z-score associated with a 95% confidence interval, from the standard normal tables, is;

1.96

substituting these values into the formula;

24 ± 1.96*(0.4/7)

= 24 ± 0.112

from the alternatives given, choice d is closest to the above expression;