Are points of inflection considered critical points?

M

math

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what are critical points of y = (e^x )(x^3)

y' = xe^x(x^3 + 3x^2)
i set this equal to 0 and after using second derivative test i found a local max and an absolute min.

do i need to include anything else? (any other critical points ie- are points of inflection considered critical points?)


thanks
 
It's a bit difficult to say, since you didn't share your results.

Are you SURE you found a Local Maximum?
 
hi, im really sorry the problem was actually y = (e^x)(x^2). identify critical points

y' = e^x ( x^2 + 2x) = 0
x = -2 x=0
(-2, 4e^-2) is absolute max (0, 0) is local minimum
Are these the only two critical points, or would i also have to state the points of inflection, which i think would be:
y'' = 0 = e^x(x^2+4x+2)
x=-sqrt2 -2 and x = sqrt2- 2
 
Why don't you factor the derivative correctly?
\(\displaystyle \L xe^x (x + 2).\)
 
OK. y' = xe^x(x+2)
any critical points besides absolute max and local min? and if the points of inflection are critical points, are they correct?
thanks again
 
Do you think points of inflection are important? I find the Point of Inflection at \(\displaystyle \L\;(-2-\sqrt{2})\) to be very important. It's the way to start believing the thing doesn't cross the x-axis again.
 
Ok. Yes, i guess it would be a critical point then. thanks. rad2-2 would also be a critical point, right? i was confused because i thought critical points were only when y' is 0 or undefined. is this not true?
 
Critical points are whatever your book or teacher defines them to be. I think the most useful definition is "anything we need to know".

"rad2-2" is ambiguous. Maybe "rad(2) - 2" would be more clear.

Also, any endpoints or holes would be consdiered "critical" by me. Anymptotes?
 
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