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  4. The point of inflexion on the curve y=x5/3 will be

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Q. The point of inflexion on the curve $ y={{x}^{5/3}} $ will be

Rajasthan PETRajasthan PET 2004

Solution:

Given, $ y={{x}^{5/3}} $
$ \Rightarrow $ $ \frac{dy}{dx}=\frac{5}{3}{{x}^{2/3}} $
For the point of inflexion,
$ \frac{dy}{dx}=0 $
$ \Rightarrow $ $ \frac{5}{3}{{x}^{2/3}}=0 $
$ \Rightarrow $ $ x=0 $
$ \Rightarrow $ $ y=0 $ So, point is (0, 0)
Now, $ \frac{{{d}^{2}}y}{d{{x}^{2}}}=\frac{5}{3}.\frac{2}{3}{{x}^{-1/3}} $
$ {{\left( \frac{{{d}^{2}}y}{d{{x}^{2}}} \right)}_{x=0}}=0 $
Hence, point of inflexion does not exist.