How do you find all points of inflection given #y=x^5+2x^3+4#?
Inflection points at
Find the second derivative:
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To find all points of inflection for the function ( y = x^5 + 2x^3 + 4 ), follow these steps:
 Find the second derivative of the function.
 Set the second derivative equal to zero and solve for ( x ) to find potential points of inflection.
 Test each potential point of inflection by checking the sign of the second derivative around that point.
 Points where the second derivative changes sign are the points of inflection.
Let's go through these steps:

Find the second derivative of ( y ): ( y' = 5x^4 + 6x^2 ) ( y'' = 20x^3 + 12x )

Set the second derivative equal to zero and solve for ( x ): ( 20x^3 + 12x = 0 ) ( 4x(5x^2 + 3) = 0 ) ( x = 0 ) or ( x = \pm \sqrt{\frac{3}{5}} )

Test each potential point of inflection:
 For ( x = 0 ), evaluate ( y''(x) ) for values slightly less and slightly more than 0.
 For ( x = \pm \sqrt{\frac{3}{5}} ), evaluate ( y''(x) ) for values slightly less and slightly more than ( \pm \sqrt{\frac{3}{5}} ).

Determine the sign of ( y''(x) ) around each potential point of inflection. Points where the sign changes indicate points of inflection.
Therefore, the points of inflection for the function ( y = x^5 + 2x^3 + 4 ) occur at ( x = 0 ) and ( x = \pm \sqrt{\frac{3}{5}} ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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