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Multiple Choice Questions

271.

The point on the curve x² = 2y which is nearest to the point (0, 5) is

$left parenthesis 2 square root of 2 comma space space 4 right parenthesis$
$left parenthesis 2 square root of 2 comma space space 0 right parenthesis$
(0, 0)
(0, 0)

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Short Answer Type

272.

Determine two positive numbers whose sum is 15 and the sum of whose, squares is minimum.

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273.
Amongst all pairs of positive numbers with product 64, find those whose sum is the least.

Let $straight x comma space 64 over straight x$ be the two numbers
$Let space straight S space equals space straight x plus 64 over straight x space space space space rightwards double arrow space space space dS over dx space equals space 1 minus 64 over straight x squared$
$dS over dx space equals 0 space space gives space us space 1 minus 64 over straight x squared space equals space 0 comma$
$therefore space space space space straight x squared minus 64 space equals space 0 space space space space space rightwards double arrow space space space space straight x squared space equals space 64 comma space space space space space space rightwards double arrow space space straight x space equals space 8 comma space space minus 8 Rejecting space straight x space equals space minus 8 space left parenthesis as space numbers space are space plus ve right parenthesis space we space get space straight x space equals space 8 space space space space space space space space space space space space space space space space space space space space space fraction numerator straight d squared straight S over denominator dx squared end fraction space equals space 128 over straight x cubed At space space space space straight x space equals 8 comma space space space space space fraction numerator straight d squared straight S over denominator dx squared end fraction space equals space fraction numerator 128 over denominator left parenthesis 8 right parenthesis cubed end fraction space equals space 1 fourth greater than 0 therefore space space space space straight S space is space minimum space when space straight x space equals space 8 therefore space space space space numbers space are space 8 comma space 64 over 8 space space or space space 8 comma space 8$

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274.

Determine two positive numbers whose sum is 30 and whose product is the maximum.

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275.

Find two numbers whose sum is 24 and whose product is as large as possible.

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276. Find two positive numbers whose sum is 16 and sum of whose cubes is minimum.

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277. Find two positive numbers a and y such that x + y = 40 and x y³ is maximum.

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278.

Find two positive numbers x and y such that x + y = 60 and xy³ is maximum.

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279.

Find two positive numbers a and y such that their sum is 35 and the product x²y⁵ is a maximum.

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280.