# What are the points of inflection of #f(x)=x^2-x^(1/2) #?

No Points of inflection

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The user wants accurate and comprehensive answers without irrelevant information or introductions.To find the points of inflection of ( f(x) = x^2 - x^{1/2} ), we first need to find the second derivative and then determine where it equals zero or is undefined. Let's start by finding the first and second derivatives of ( f(x) ):

First derivative: ( f'(x) = 2x - \frac{1}{2}x^{-\frac{1}{2}} ) Second derivative: ( f''(x) = 2 + \frac{1}{4}x^{-\frac{3}{2}} )

Now, let's set the second derivative equal to zero and solve for ( x ):

( 2 + \frac{1}{4}x^{-\frac{3}{2}} = 0 )

Solving for ( x ):

( \frac{1}{4}x^{-\frac{3}{2}} = -2 ) ( x^{-\frac{3}{2}} = -8 ) ( \frac{1}{x^{\frac{3}{2}}} = -8 ) ( x^{\frac{3}{2}} = -\frac{1}{8} ) ( x = \left(-\frac{1}{8}\right)^{\frac{2}{3}} ) ( x = -\frac{1}{4} )

So, the point of inflection of ( f(x) = x^2 - x^{1/2} ) is at ( x = -\frac{1}{4} ).

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To find the points of inflection of ( f(x) = x^2 - \sqrt{x} ), follow these steps:

- Calculate the second derivative, ( f''(x) ).
- Set ( f''(x) = 0 ) and solve for the values of ( x ).
- Determine the concavity of ( f(x) ) around the critical points found in step 2.
- Verify whether the concavity changes at those critical points.
- The points where the concavity changes are the points of inflection.

By following these steps, you can determine the points of inflection for the given function.

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