Inflection points and concavity

dylanfehr

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This is looking for the concavity and inflection points. f(x) = x^4 − 32x^2 + 6

I did the second derivative test and plugged them into f(x) to get y and I an having trouble. I need help with the points of inflection and intervals of concave up and down. Thank you
I need both inflection points and where the function is concave up and down

Can someone work this for me please
 
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Start with where the derivative equals zero. Those are your points of interest. Graph it for a better picture.

f'(x)=4x^3-64x=0. So your maxima and minima are at x=-4, x=4, and x=0

You can sub the x values back in at each point to find your y values and determine concavity.
 
Start with where the derivative equals zero. Those are your points of interest. Graph it for a better picture.

f'(x)=4x^3-64x=0. So your maxima and minima are at x=-4, x=4, and x=0

You can sub the x values back in at each point to find your y values and determine concavity.

While this is good to know, it is not part of what is asked for.

Skip the above and go right on to the second derivative:


f"(x) = 12x^2 - 64

12x^2 - 64 = 0

4(3x^2 - 16) = 0

\(\displaystyle 3x^2 - 16 \ = \ 0\)

\(\displaystyle 3x^2 \ = \ 16\)

\(\displaystyle x^2 \ = \ \dfrac{16}{3}\)

\(\displaystyle x \ = \ \pm\dfrac{4}{\sqrt{3}}\)

\(\displaystyle x \ = \ \pm\dfrac{4\sqrt{3}}{3}\)


At these two x-values are where there are potential inflection points. A suggestion is to test
convenient integers in the second derivative. There will be three in total, that fall on either side
of these two x-values, that will be substituted into the second derivative to determine their sign.


f"(test value) < 0 means concave down for that region

f"(test value) > 0 means concave up for that region


If the concavity changes around a particular x-value, then there is an inflection point there.

Substitute that x-value into the original function to get its y-value.

You should be able to answer the intervals of concavity from the information on the test values above.
 
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