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

y'' = 0 at an inflexion point

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

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

Furthermore, this indicates an inflexion point if y''' is not 0.

It is not zero here, y'''.

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

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To find the points of inflection of ( f(x) = x^2 - \frac{3}{x^3} ), we first need to find its second derivative and then determine where this second derivative changes sign.

The first derivative of ( f(x) ) is ( f'(x) = 2x + \frac{9}{x^4} ).

The second derivative of ( f(x) ) is ( f''(x) = 2 - \frac{36}{x^5} ).

To find where ( f''(x) ) changes sign, we set ( f''(x) = 0 ) and solve for ( x ):

[ 2 - \frac{36}{x^5} = 0 ] [ \frac{36}{x^5} = 2 ] [ x^5 = 18 ] [ x = \sqrt[5]{18} ]

Since ( f''(x) ) changes sign at ( x = \sqrt[5]{18} ), this point is a point of inflection for ( f(x) = x^2 - \frac{3}{x^3} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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