Application of Derivatives

Question

A particle moves along the curve 6y = x³ + 2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.

Solution

Here, 6y = x³ + 2. ...(1)
Differentiating both sides, w.r.t 't'
$6 dy over dt space equals space 3 straight x squared dx over dt$
But $dy over dt space equals space 8 dx over dt$ (given)
$therefore space space space space space space space 6 open parentheses 8 dx over dt close parentheses space equals space 3 straight x squared dx over dt space space space space space rightwards double arrow space space space space 16 space equals space straight x squared space space space space space space space space space space rightwards double arrow space space space space straight x space equals space 4 comma space space space minus 4$
$When space space space space straight x space equals space 4 space from space left parenthesis 1 right parenthesis comma space we space get comma space space space space space straight y space equals space fraction numerator left parenthesis 4 right parenthesis cubed plus 2 over denominator 6 end fraction space equals space 66 over 6 space equals space 11 When space space space straight x space equals space minus 4 space space from space left parenthesis 1 right parenthesis comma space we space get comma space space space straight y space equals space fraction numerator left parenthesis negative 4 right parenthesis cubed plus 2 over denominator 6 end fraction space equals space fraction numerator negative 62 over denominator 6 end fraction space equals space minus 31 over 3 therefore space space space space required space points space on space the space curve space are space left parenthesis 4 comma space 11 right parenthesis comma space space open parentheses negative 4 comma space space space space space space minus 31 over 3 close parentheses.$

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