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  4. On the curve x3=12y, the abscissa changes at a faster rate than the ordinate. Then, x belongs to the interval

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Q. On the curve $ {{x}^{3}}=12y, $ the abscissa changes at a faster rate than the ordinate. Then, $ x $ belongs to the interval

JamiaJamia 2011

Solution:

Given, $ {{x}^{3}}=12y $ $ \Rightarrow $ $ 3{{x}^{2}}\frac{dx}{dy}=12 $ $ \Rightarrow $ $ \frac{dx}{dy}=\frac{4}{{{x}^{2}}} $ But it is given $ \left| \frac{dy}{dx} \right|>1 $ $ \therefore $ $ \frac{4}{{{x}^{2}}}>1 $ $ \Rightarrow $ $ {{x}^{2}}-4<0 $ $ \Rightarrow $ $ -2