Application of Derivatives

Find the points on the curve which are nearest to the point (0, 5). - Wired Faculty

Question

Find the points on the curve $straight y equals straight x squared over 4$ which are nearest to the point (0, 5).

Solution

Let (x, y) be the point on $straight y equals straight x squared over 4$ which is nearest to the point (0, 5).
$because space space space space left parenthesis straight x comma space straight y right parenthesis space lies space on space the space curve space straight y space equals space straight x squared over 4 space space space space rightwards double arrow space space space point space is space open parentheses straight x comma space straight x squared over 4 close parentheses$
Let d be the distance between (0, 5) and $open parentheses straight x comma space space straight x squared over 4 close parentheses$
$therefore space space space space straight d space equals space square root of left parenthesis straight x minus 0 right parenthesis squared plus open parentheses straight x squared over 4 minus 5 close parentheses squared end root space space space space space space space space space space space space space rightwards double arrow space space space space straight d squared space equals straight x squared plus open parentheses straight x squared over 4 minus 5 close parentheses squared Put space straight d squared equals space straight D comma space space space space space therefore space space space straight D space equals space straight x squared plus open parentheses straight x squared over 4 minus 5 close parentheses squared$
Now d is maximum or minimum when D is maximum or minimum.
$dD over dx space equals 2 straight x plus 2 open parentheses straight x squared over 4 minus 5 close parentheses. space space space fraction numerator 2 straight x over denominator 4 end fraction equals space space 2 straight x plus straight x open parentheses straight x squared over 4 minus 5 close parentheses dD over dx space equals space 0 space give space us space 2 straight x plus straight x open parentheses straight x squared over 4 minus 5 close parentheses space equals space 0 space space space space rightwards double arrow space space space space 8 straight x plus straight x cubed minus 20 straight x space equals space 0 therefore space space space space space straight x cubed minus 12 straight x space equals space 0 space space or space space space straight x left parenthesis straight x squared minus 12 right parenthesis space equals space 0 space space space space space space space rightwards double arrow space space straight x left parenthesis straight x minus square root of 12 right parenthesis thin space left parenthesis straight x plus square root of 12 right parenthesis space equals space 0 therefore space space space space space straight x equals space space 0 comma space space space space 2 square root of 3 comma space space minus 2 square root of 3 space space space space space space space space fraction numerator straight d squared straight D over denominator dx squared end fraction space equals space 2 plus straight x. fraction numerator 2 straight x over denominator 4 end fraction plus open parentheses straight x squared over 4 minus 5 close parentheses.1 space space equals 2 plus straight x squared over 2 plus straight x squared over 4 minus 5 space equals space fraction numerator 3 straight x squared over denominator 4 end fraction minus 3$
$At space straight x space equals space 0 comma space space space fraction numerator straight d squared straight D over denominator dx squared end fraction equals space 0 minus 3 space equals space minus 3 space less than space 0 therefore space space space space straight D space is space maximum space when space straight x space equals space 0 We space are space not space interested space in space this space case At space straight x space equals space 2 square root of 3 comma space space fraction numerator straight d squared straight D over denominator dx squared end fraction space equals 3 over 4. space 12 space minus 3 space equals space 9 minus 3 space equals space 6 greater than 0 therefore space space space space space straight D space is space minimum space when space straight x space equals space 2 square root of 3 comma space space space straight y space equals space 12 over 4 space equals space 3 therefore space space space space space point space is space left parenthesis 2 square root of 3 comma space 3 right parenthesis$
Again at $straight x equals negative 2 square root of 3 comma space space fraction numerator straight d squared straight D over denominator dx squared end fraction space equals space 3 over 4.12 space minus space 3 space equals space 6 space greater than space 0$
$therefore space space space space straight D space is space minimum space when space straight x space equals 2 square root of 3 comma space space space straight y space equals space 12 over 4 space equals space 3 space space space space space space space rightwards double arrow space space space space points space is space left parenthesis negative 2 square root of 3 comma space 3 right parenthesis$

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Application Of Derivatives

Find the points on the curve $straight y equals straight x squared over 4$ which are nearest to the point (0, 5).

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