onegirl
For what intervals is f(x) = 2x^4 – 4x^2 + 6 increasing?
A. The graph is increasing on the intervals (-1, 0) and (1, ∞).
B. The graph is increasing on the interval (1, ∞).
C. The graph is increasing on the interval (-1, 0).
D. The graph is increasing on the intervals ( -∞ , -1) and (0, 1).
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bfsgd
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You can differentiate the function and it is increasing if f ' (x)>0.
onegirl
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huh?
onegirl
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@electrokid can u help?
bfsgd
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Differentiating f(x) we get \[f'(x)=8x^3 - 8x=8x(x^2-1)\] That is positive for all x>1.
onegirl
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ok
onegirl
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so what do i do next?
bfsgd
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That's it :)
onegirl
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ohh so it will be D?
bfsgd
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No...D includes negative numbers. You need the one thats only x>1.
electrokid
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|dw:1364917996121:dw|
get the critical points
you see the four intervals?
take an arbitrary number from each interval and check the signs of the derivative
onegirl
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Ohh so its B
bfsgd
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Yep :)
electrokid
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nope
onegirl
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what do you mean nope?
onegirl
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so its wrong?
bfsgd
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Its right....
electrokid
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yep. B is not the answer
electrokid
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check the interval (-1,0)
electrokid
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in math, there are no shortcuts. follow the "yellow brick road"
onegirl
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ok
electrokid
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do you see the mistake?
onegirl
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yea
electrokid
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so, what is the answer?
onegirl
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i got C
electrokid
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follow the steps.
did you use the number line that I drew?
onegirl
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yea hold on
onegirl
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so the critical points are -1, 0 and 1
onegirl
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okay
electrokid
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sorry. yes. you are correct. "3" points, -1, 0 and 1
onegirl
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lol ok
electrokid
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|dw:1364919097503:dw|
electrokid
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interval (-inf, -1)
plug in x=-2 and find f'(x)
onegirl
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plug in -2 into the problem?
electrokid
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yep
onegirl
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into (x - 2)^2?
electrokid
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into the entire thing
f'(x)
onegirl
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ok
onegirl
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idk i got 16?
onegirl
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and the derivative i got 0
electrokid
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\[f'(-2)=8(-2)[(-2)^2-1]=?\]
onegirl
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ohh i got -48
electrokid
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good.|dw:1364919435696:dw|
now, we check interval (-1,0)
find \[f'(-0.5)\]
onegirl
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okay i get 0
electrokid
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nope
try again.
onegirl
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you mean find the derivative of -0.5 ?
electrokid
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derivative "at" x=-0.5
onegirl
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ohhh ok
electrokid
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yep. what did you get?
onegirl
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i'm still getting 0 :/
electrokid
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\[f'(-0.5)=8(-0.5)[(-0.5)^2-1]=?\]
onegirl
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ok i get 3
electrokid
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good.|dw:1364919843444:dw|
now we check interval (0,1)
lets take x = 0.5
similarly, find \[f'(0.5)\]
onegirl
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okay
onegirl
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i get -3
electrokid
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great
|dw:1364919912982:dw|
now the final interval (1,+infty)
we take x= 2
\[f'(2)=?\]
onegirl
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ok i get 48
electrokid
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perfect.
|dw:1364919991818:dw|
now,
f(x) is increasing when f'(x) >0 (i.e., f'(x) is positive)
and
f(x) is decreasing when f'(x) < 0 (i.e., f'(x) is negative)
electrokid
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so, in what intervals do you see f'(x) positive?
onegirl
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ok in 0,1 and 1, + infi
electrokid
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great.
which option says that?
onegirl
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A!
electrokid
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and that is your answer
yay
onegirl
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thanks :)
electrokid
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follow the same procedure for the other one,
onegirl
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ok