How do you find points of inflection and determine the intervals of concavity given #y=2x^44x^2+1#?
inflection points
concave up
concave down
Using the second derivative test, let's:
Determine the relative highest and lowest points:
Locate the inflection points:
Concavity intervals:
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To find points of inflection and determine the intervals of concavity for the function ( y = 2x^4  4x^2 + 1 ), follow these steps:
 Find the second derivative of the function.
 Set the second derivative equal to zero and solve for ( x ) to find the potential points of inflection.
 Test the intervals between these points of inflection using the second derivative test to determine the concavity of the function.
Let's go through each step:

Find the second derivative of the function ( y = 2x^4  4x^2 + 1 ): [ y' = 8x^3  8x ] [ y'' = 24x^2  8 ]

Set the second derivative equal to zero and solve for ( x ) to find potential points of inflection: [ 24x^2  8 = 0 ] [ 24x^2 = 8 ] [ x^2 = \frac{8}{24} = \frac{1}{3} ] [ x = \pm \sqrt{\frac{1}{3}} ]

Test the intervals between these points of inflection:
 Choose a value of ( x ) in each interval and plug it into the second derivative.
 If ( y'' > 0 ), the function is concave up in that interval.
 If ( y'' < 0 ), the function is concave down in that interval.
Choose ( x = 0 ) for the interval ( (\infty, \sqrt{\frac{1}{3}}) ): [ y'' = 24(0)^2  8 = 8 < 0 ] So, the function is concave down in this interval.
Choose ( x = \frac{1}{2} ) for the interval ( (\sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}}) ): [ y'' = 24\left(\frac{1}{2}\right)^2  8 = 6  8 = 2 < 0 ] So, the function is concave down in this interval.
Choose ( x = 1 ) for the interval ( (\sqrt{\frac{1}{3}}, \infty) ): [ y'' = 24(1)^2  8 = 16 > 0 ] So, the function is concave up in this interval.
Therefore, the points of inflection are ( (\sqrt{\frac{1}{3}}, \frac{1}{2}) ) and ( (\sqrt{\frac{1}{3}}, \frac{1}{2}) ), and the intervals of concavity are ( (\infty, \sqrt{\frac{1}{3}}) ) and ( (\sqrt{\frac{1}{3}}, \infty) ), where the function is concave down, and ( (\sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}}) ), where the function is concave up.
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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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