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Application of Derivatives

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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.

Let one number = x
$therefore space space space other space number space equals space 15 minus straight x Let space space straight S space equals space straight x squared plus left parenthesis 15 minus straight x right parenthesis squared space equals space 2 straight x squared minus 30 straight x plus 225 space space space space space dS over dx space equals space 4 straight x minus 30 space space space space space dS over dx space equals space 0 space space space gives space us space space 4 straight x minus 30 space equals space 0 space space space space space rightwards double arrow space space space straight x space equals space 15 over 2 space space space fraction numerator straight d squared straight S over denominator dx squared end fraction space equals space 4 At space straight x space equals space 15 over 2. space fraction numerator straight d squared straight S over denominator dx squared end fraction space equals space 4 comma space space space which space is space plus ve therefore space space space space space straight S space is space minimumn space when space space straight x space equals space 15 over 2 therefore space space space space one space number space equals space 15 over 2 space and space other space number space equals space 15 minus 15 over 2 space equals space 15 over 2$

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273.

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

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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.