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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)

left parenthesis 2 square root of 2 comma space space 4 right parenthesis

Let (x, y) be the point on $straight x squared space equals space 2 straight y$ which is nearest to the point (0, 5).
$because space space space space left parenthesis straight x comma space straight y right parenthesis space lies space on space the space curve space straight y space equals space straight x squared over 2 therefore space space space space space space point space is space open parentheses straight x comma space space straight x squared over 2 close parentheses Let space straight d space be space the space distance space between space left parenthesis 0 comma space 5 right parenthesis space and space open parentheses straight x comma space straight x squared over 2 close parentheses. therefore space space space space space straight d space equals space square root of left parenthesis straight x minus 0 right parenthesis squared plus open parentheses straight x squared over 2 minus 5 close parentheses end root space space space space rightwards double arrow space space space space space straight d squared space equals straight x squared plus open parentheses straight x squared over 2 minus 5 close parentheses squared Put space straight d squared space equals space straight D. space space space space space space therefore space space space space space straight D space equals space straight x squared plus space open parentheses straight x squared over 2 minus 5 close parentheses squared$

   $dD over dx space equals space 2 straight x plus 2 open parentheses straight x squared over 2 minus 5 close parentheses. space fraction numerator 2 straight x over denominator 2 end fraction space equals 2 straight x plus straight x left parenthesis straight x squared minus 10 right parenthesis$

   $equals 2 straight x plus straight x cubed minus 10 straight x equals space space straight x cubed minus 8 straight x$

   $dD over dx space equals 0 space space space give space us space straight x cubed minus 8 straight x space equals space 0$

$therefore space space space space straight x left parenthesis straight x squared minus 8 right parenthesis space equals space 0 space space space space space rightwards double arrow space space space space straight x space equals space 0 comma space space space plus-or-minus square root of 8 space equals space 0 comma space plus-or-minus 2 square root of 2 space space space space space space space space space space fraction numerator straight d squared straight D over denominator dx squared end fraction space equals space 3 straight x squared minus 8 At space space straight x space equals space 0 comma space space space fraction numerator straight d squared straight D over denominator dx squared end fraction space equals space 0 minus 8 space equals space space minus 8 space less than 0 space therefore space space space space space straight D space is space maximum space when space straight x space equals space 0$

We are not interested in this case.
$At space space straight x space equals space 2 square root of 2 comma space space space fraction numerator straight d squared straight D over denominator dx squared end fraction space equals space 3 space cross times space 8 space minus space 8 space space equals space 16 space greater than space 0 therefore space space space straight D space is space minimum space when space straight x space equals space 2 square root of 2 comma space space space straight y space equals space 8 over 2 space equals space 4 therefore space space space point space is space left parenthesis 2 square root of 2 comma space space 4 right parenthesis$

$At space space straight x space equals space minus 2 square root of 2 comma space space space space space fraction numerator straight d squared straight D over denominator dx squared end fraction space equals space 3 space cross times 8 space minus space 8 space equals space 16 space greater than space 0 space therefore space space space space straight D space is space minimum space when space straight x space equals space minus 2 square root of 2 comma space space space straight y space equals space 8 over 2 space equals space 4 therefore space space space space point space is space left parenthesis negative 2 square root of 2 comma space space 4 right parenthesis therefore space space space left parenthesis straight A right parenthesis space is space correct space answer.$

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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.

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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.