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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.

Let one number = x.
$therefore space space space space space other space number space equals space 30 minus straight x$
$Let space straight P space equals space straight x left parenthesis 30 minus straight x right parenthesis space equals space 30 straight x minus straight x squared space space space space space space space space space space space space space space space rightwards double arrow space space dP over dx equals 30 minus 2 straight x$
$dP over dx space equals space 0 space space gives space us space 30 minus 2 straight x space equals space 0 comma space space space space space space space space space space space space space rightwards double arrow space space straight x space equals space 15 fraction numerator straight d squared straight P over denominator dx squared end fraction space equals space minus 2$
$At space straight x space equals space 15 comma space space space space fraction numerator straight d squared straight P over denominator dx squared end fraction space equals negative 2 space less than space 0 therefore space space space straight P space is space maximum space when space straight x space equals space 15 space space space space rightwards double arrow space space space numbers space are space 15 comma space 30 comma space minus 15 space space straight i. straight e comma space 15 comma space 15$