# What are the points of inflection, if any, of #f(x)= x^5 -7 x^3- x^2-2 #?

We can't easily find exact locations, but there are three inflection points at roughly x = -1.4, 0, and 1.5. Graphs below.

To find inflection points we want the concavity to change from concave up to concave down, usually the second derivative is zero (or undefined) there.

Set f''(x) = 0 (since for polynomials the derivatives are never undefined):

This f''(x) = 0 equation doesn't factor or have rational roots, but does have three solutions, at about -1.42, -0.04, and 1.47. (I used tables of values to zoom in on the roots.)

The original function with its three inflection points:

// dansmath \\ strikes again!

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To find the points of inflection of ( f(x) = x^5 - 7x^3 - x^2 - 2 ), we first need to find the second derivative, ( f''(x) ). Then, we solve for the values of ( x ) where ( f''(x) = 0 ) or ( f''(x) ) is undefined. Finally, we determine if the concavity changes sign at those points.

( f'(x) = 5x^4 - 21x^2 - 2x ) ( f''(x) = 20x^3 - 42x - 2 )

To find the points of inflection, set ( f''(x) = 0 ) and solve for ( x ): ( 20x^3 - 42x - 2 = 0 )

This equation may not have easy-to-find solutions, so numerical methods or approximation may be necessary to find the exact points of inflection. Once the solutions for ( x ) are found, the corresponding ( y ) values can be determined by substituting the ( x ) values into the original function ( f(x) ).

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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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