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Home Q & A Board Science-Related Homework Help Mathematics Calculus
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slimes slimes
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Posts: 162
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3 years ago
for the function f(x)= x^4 - 6x^2 - 5, find the point of inflection and the intervals of concavity
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wrote...
#1
Educator
3 years ago
Hi slimes

See this tutorial:

Point of inflection:



Concavity test:
slimes Author
wrote...
#2
3 years ago
Hey bio_man thanks, I'm still confused with this question, I'm on the points of inflection part, but I ended up getting x = 1, not sure if that's correct.
wrote...
#3
Educator
3 years ago
\(f(x)=x^4-6x^2-5\)

\(f'(x)=4x^3-12x\)

\(f''(x)=12x^2-12\)

Set \(f''(x)=0\) and solve for x:

\(0=12x^2-12\)

\(12=12x^2\)

\(\frac{12}{12}=x^2\)

\(1=x^2\)

\(x=\pm 1\)

Find out what happens between -∞ to -1
Find out what happens between -1 and 1
Find out what happens between 1 and ∞

Watch the first video again to see how to conclude from there...
slimes Author
wrote...
#4
3 years ago
Hi bio_man thank you so much, but if I'm completely honest, we barley learned this topic and we are already being tested on this!! I have been watching a bunch of vides to help me answer these questions but not much use. But so far this video has been very helpful because I did get the answer that you had just showed me above. Would the intervals of concavity be -12 and 0?
wrote...
#5
Educator
3 years ago Edited: 3 years ago, bio_man
Cool, but can't be -12 because the critical points are -1 and 1 only. Like I said above, select a random point between those intervals and see what the sign is, I.e. whether it be negative or positive. Create for me a table of your findings, and from there we can conclude it together.
slimes Author
wrote...
#6
3 years ago
I found that 0.5 gives -9 (-) but tried everything else between 1 and -1 and they were all negative when I plugged it into the 2nd derivative
wrote...
#7
Educator
3 years ago
\(f''(x)=12x^2-12\)

Let's try -2, 0, and 2

\(f''(-2)=12\left(-2\right)^2-12=36\) [positive]

\(f''(0)=12\left(0\right)^2-12=-12\) [negative]

\(f''(2)=12\left(2\right)^2-12=36\) [positive]

Now try concluding...
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