Consider the curve defined by x 2 + 2 y 2 + 4 β x y = K with K &gt; 0 and where β is a (sufficiently small) parameter. Assuming that the above can be used to define a function x = G ( y ), use implicit differentiation to find out what is the largest value that x can take on along this curve.My attempt: Since x = G ( y ), the largest value of x is obtained when G ( y ) is maximized, or when G ′ ( y ) = 0. However, the given equation of the curve isn't G ( y ) itself. How do I express the curve equation as G ( y ), or is there another way of finding the largest value of x along this curve?

Question

Consider the curve defined by       x    2    +  2      y    2    +  4  β  x  y  =  K with   K  &amp;gt;  0 and where   β is a (sufficiently small) parameter. Assuming that the above can be used to define a function   x  =  G  (  y  ), use implicit differentiation to find out what is the largest value that   x can take on along this curve.My attempt: Since   x  =  G  (  y  ), the largest value of   x is obtained when   G  (  y  ) is maximized, or when       G    ′    (  y  )  =  0. However, the given equation of the curve isn&#039;t   G  (  y  ) itself. How do I express the curve equation as   G  (  y  ), or is there another way of finding the largest value of   x along this curve?

Dillon Levy · Accepted Answer

If you differentiate the given expression above, say   F  (  x  ,  y  )  −  K  =  0, with respect to   y, considering   x  =  x  (  y  ), you would have:  2    x  (  y  )        x    ′    (  y  )  +  4  y  +  4  β    (      x    ′    (  y  )    y  +  x  (  y  )  )  =  0  ,where use has been made of the chain rule.

Consider the curve defined by x^2+2y^2+4beta xy=K with K>0 and where beta is a (sufficiently small) parameter. Assuming that the above can be used to define a function x=G(y), use implicit differentiation to find out what is the largest value that x can take on along this curve.

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