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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.

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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.

Let two numbers be x and 24 - x.
Let P = x(24 - x) = 24x - x^{2 $rightwards double arrow space space space space dP over dx space equals space 24 minus 2 straight x.$
$dP over dx equals space 0 space space space gives space us space space 24 minus 2 straight x space equals space 0 comma space space space space space space rightwards double arrow space space straight x space equals space 12 fraction numerator straight d squared straight P over denominator dx squared end fraction equals negative 2 At space space space straight x space equals space 12 comma space space space space fraction numerator straight d squared straight P over denominator dx squared end fraction space equals negative 2 space less than space 0 therefore space space space straight P space is space maximum space space when space space straight x space equals space 12 space space space space space space space rightwards double arrow space space space numbers space are space 12 comma space 24 minus 12 space straight i. straight e. comma space 12 comma space 12$}

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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.