Explanation Given information: The function is g ( x , y ) = 11 x y + 14 V x + 15 V y . Formula used: Second derivative test: Suppose that f ( x , y ) is a function and ( a , b ) is a point at which ∂ f ∂ x ( a , b ) = 0 and ∂ f ∂ y ( a , b ) = 0 , and let D ( x , y ) = ∂ 2 f ∂ x 2 × ∂ 2 f ∂ y 2 − ( ∂ 2 f ∂ x ∂ y ) 2 . It has four different cases to determine a relative maximum and minimum. If D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) > 0 , then f ( x , y ) has a relative minimum at ( a , b ) . If D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) < 0 , then f ( x , y ) has a relative maximum at ( a , b ) . If D ( a , b ) < 0 , then f ( x , y ) has neither a relative maximum nor a relative minimum at ( a , b ) . If D ( a , b ) = 0 , then no conclusion can be drawn from this test. Proof: By the second derivative test, a relative minimum occurs at x = a and y = b when ∂ 2 f ∂ x 2 ( a , b ) > 0 and D ( a , b ) > 0 . Therefore, find partial derivatives ∂ 2 f ∂ x 2 , ∂ 2 f ∂ y 2 and ∂ 2 f ∂ x ∂ y . Differentiate the given function with respect to x . ∂ g ∂ x = ∂ ∂ x ( 11 x y + 14 V x + 15 V y ) ⇒ ∂ g ∂ x = 11 y − 14 V x 2 . Again, differentiate with respect to x . ∂ ∂ x ( ∂ g ∂ x ) = ∂ ∂ x ( 11 y − 14 V x 2 ) ⇒ ∂ 2 g ∂ x 2 = 28 V x 3 . Now, differentiate given function with respect to y . ∂ g ∂ y = ∂ ∂ y ( 11 x y + 14 V x + 15 V y ) ⇒ ∂ g ∂ y = 11 x − 15 V y 2

15th Edition

ISBN: 9780137590896

Author: Larry J. Goldstein; David C. Lay; David I. Schneider; Nakhle H. Asmar; William Edward Tavernetti

Publisher: Pearson Education (US)

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 7.3, Problem 2CYU

To determine

To prove: Relative minimum of the function g(x, y)=11xy+14Vx+15Vy actually occurs at x=56 and y=60 by using the second derivative test where V=147,840 cubic feet.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7.3, Problem 1CYU

Chapter 7.3, Problem 1E

Students have asked these similar questions

Write an equation for the function graphed below 5+ 4 - -7 -6 -5 -4 -3 -2 -1 y = 3. 2 1 + 1 2 3 4 5 6 7 -1 -3 -4 5 -5+ a

Approximate graphically the radius and height of a cylindrical container with volume 50 cubic inches and lateral surface area 75 square inches. h 2лr The radius is in and the height is in. (Round to the nearest hundredth.) h Volume of a cylinder = r²h Lateral area of a cylinder = 2лrh

Find the derivative of the following function. -8e5x y= 9x+2

Chapter 7 Solutions

Calculus & Its Applications

Ch. 7.1 - Let f(x,y,z)=x2+y(xz)4. Compute f(3,5,2).Ch. 7.1 - Prob. 2CYU Ch. 7.1 - Let f(x,y)=x23xyy2. Compute f(5,0),f(5,2) and...Ch. 7.1 - Prob. 2E Ch. 7.1 - Let g(x,y,z)=x/(yz). Compute g(1,1),g(0,1) and...Ch. 7.1 - Let f(x,y,z)=x2ey2+z2. Compute f(1,1,1) and...Ch. 7.1 - Let f(x,y)=xy. Show that f(2+h,3)f(2,3)=3h.Ch. 7.1 - Let f(x,y)=xy. Show that f(2,3+k)f(2,3)=2k.Ch. 7.1 - Cost Find a formula C(x,y,z) that gives the cost...Ch. 7.1 - Cost Find a formula C(x,y,z) that gives the cost...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

To prove: Relative minimum of the function g ( x , y ) = 11 x y + 14 V x + 15 V y actually occurs at x = 56 and y = 60 by using the second derivative test where V = 147 , 840 cubic feet.

Solution Summary: The author demonstrates the relative minimum of the function g(x,y)=11xy+14Vx+

Concept explainers

Videos

To prove: Relative minimum of the function g(x, y)=11xy+14Vx+15Vy actually occurs at x=56 and y=60 by using the second derivative test where V=147,840 cubic feet.

Want to see the full answer?

Chapter 7 Solutions