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Problem 1

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=1-2 x^{2}-3 y^{2}\)

Expert verified

The critical point of the function \(f(x, y) = 1-2x^2-3y^2\) is (0, 0), found by setting the partial derivatives \(\frac{\partial f}{\partial x} = -4x\) and \(\frac{\partial f}{\partial y} = -6y\) equal to 0. Using the second derivative test with the Hessian matrix determinant, \(D = 24\), it is determined that (0, 0) is a relative maximum. Therefore, the relative extrema of the function are the relative maximum (0, 0) with a function value of f(0, 0) = 1.

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Chapter 12

Postal regulations specify that the combined length and girth of a parcel sent by parcel post may not exceed 130 in. Find the dimensions of the rectangular package that would have the greatest possible volume under these regulations. Hint: Let the dimensions of the box be \(x^{\prime \prime}\) by $y^{\prime \prime}\( by \)z^{\prime \prime}\( (see the figure below). Then, \)2 x+2 z+y=130$, and the volume \(V=x y z\). Show that $$V=f(x, z)=130 x z-2 x^{2} z-2 x z^{2}$$ Maximize \(f(x, z)\)

Chapter 12

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(h\) is a function of \(x\) and \(y\), then there are functions \(f\) and \(g\) of one variable such that $$h(x, y)=f(x)+g(y)$$

Chapter 12

Find the first partial derivatives of the function. \(g(u, v, w)=\frac{2 u w w}{u^{2}+v^{2}+w^{2}}\)

Chapter 12

Boor An empirical formula by E. F. Dubois relates the surface area \(S\) of a human body (in square meters) to its weight \(W\) (in kilograms) and its height \(H\) (in centimeters). The formula, given by $$S=0.007184 W^{0.425} H^{0.725}$$ is used by physiologists in metabolism studies. a. Find the domain of the function \(S\). b. What is the surface area of a human body that weighs \(70 \mathrm{~kg}\) and has a height of \(178 \mathrm{~cm}\) ?

Chapter 12

Find the first partial derivatives of the function. \(f(x, y)=x \sqrt{1+y^{2}}\)

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