quiz 9 - requires respondus lockdown browser + we\nstarted oct 20 at 9:55pm\nquiz instructions\naccess code…

quiz 9 - requires respondus lockdown browser + we\nstarted oct 20 at 9:55pm\nquiz instructions\naccess code: start\ntimed: 25 minutes\nnumber of attempts: 1\nlockdown browser and respondus monitor required,\nthis quiz covers sections through 3.7 from the textbook.\nquestion 5\nthe sum of two numbers is 8. find the numbers if the sum of their cubes is minimal.\n○ both numbers are 4.\n○ the numbers are 2 and 6.\n○ the numbers are 0 and 8.\n○ no correct answer choice is given.\n○ both numbers are 2.

quiz 9 - requires respondus lockdown browser + we\nstarted oct 20 at 9:55pm\nquiz instructions\naccess code: start\ntimed: 25 minutes\nnumber of attempts: 1\nlockdown browser and respondus monitor required,\nthis quiz covers sections through 3.7 from the textbook.\nquestion 5\nthe sum of two numbers is 8. find the numbers if the sum of their cubes is minimal.\n○ both numbers are 4.\n○ the numbers are 2 and 6.\n○ the numbers are 0 and 8.\n○ no correct answer choice is given.\n○ both numbers are 2.

Answer

Explanation:

Step1: Define variables

Let one number be (x), then the other number is (8 - x). The sum of their cubes (S=x^{3}+(8 - x)^{3}).

Step2: Expand the function

Using the formula ((a - b)^{3}=a^{3}-3a^{2}b + 3ab^{2}-b^{3}), we have (S=x^{3}+(8^{3}-3\times8^{2}x+3\times8x^{2}-x^{3})). Simplify (S = 512-192x + 24x^{2}).

Step3: Find the derivative

Differentiate (S) with respect to (x). (S^\prime=\frac{dS}{dx}=-192 + 48x).

Step4: Set the derivative equal to zero

Set (S^\prime = 0), then (-192+48x = 0). Solve for (x): (48x=192), so (x = 4).

Step5: Check the second - derivative

Differentiate (S^\prime) to get the second - derivative (S^{\prime\prime}=48>0). Since (S^{\prime\prime}>0) when (x = 4), the function (S) has a minimum at (x = 4).

Answer:

Both numbers are 4.