What are the points of inflection, if any, of #f(x)=x^2 - 3/x^3 #?

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Answer 1 · 2016-12-15T11:31:22

#(18^(1/5), 18^(2/5)-3/18^(3/5))=(1.783, 2.648)#, nearly.. Look for this point in the inserted graph, at which the tangent is crossing the curve.

At a point of inflexion, y'' = 0

graph{x^2-3/x^3 [-2, 5, -5, 5]}

#y=x^2-3/x^3#

#y'=2x+9/x^4#

y''=2-36/x^5=0, at x =18^(1/5#

Further, if y''' is not 0 here, this gives a point of inflexion.

Here, y''' is not 0.

So, (18^(1/5), 18^(2/5-3/18^(3/5)=(1.783, 2.648)#, nearly, is the point of

inflexion (POI).

The scale on the x-axis is changed to disclose #tangent-crossing-

curve# at POI.s