NathanJHW
  • NathanJHW
The figure at below is the graph of f ′′ (x), the second derivative of a function f (x). The domain of the function f (x) is all real numbers, and the graph shows f ′′ (x) for −2.6≤ x ≤3.6.
Mathematics
  • Stacey Warren - Expert brainly.com
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jamiebookeater
  • jamiebookeater
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NathanJHW
  • NathanJHW
1 Attachment
NathanJHW
  • NathanJHW
A. Find all values of x in the interval (−2.6, 3.6) where f ′(x) has a horizontal tangent. B. Find all values of x in the interval (−2.6, 3.6) where f (x) is concave upwards. Explain your answer. C. Suppose it is known that in the interval (−3.6, 3.6), f (x) has critical points at x =1.37, and x = −0 .97. Classify these points as relative maxima or minima of f (x). Explain your answer.
NathanJHW
  • NathanJHW
@jim_thompson5910

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jim_thompson5910
  • jim_thompson5910
what do you have so far?
NathanJHW
  • NathanJHW
Don't know where to start.
jim_thompson5910
  • jim_thompson5910
A. Find all values of x in the interval (−2.6, 3.6) where f ′(x) has a horizontal tangent.
jim_thompson5910
  • jim_thompson5910
f ' (x) has a horizontal tangent whenever the tangent slope on f ' (x) is 0 that only happens when f '' (x) = 0
jim_thompson5910
  • jim_thompson5910
It's very similar to saying "f(x) has a horizontal tangent at the point where f ' (x) = 0"
NathanJHW
  • NathanJHW
So would occur at the maximums and minimums?
jim_thompson5910
  • jim_thompson5910
no I'm saying that the horizontal tangent on f ' (x) happens at the roots of f '' (x)
NathanJHW
  • NathanJHW
So how do we find those?
jim_thompson5910
  • jim_thompson5910
what are the roots of f '' (x)
jim_thompson5910
  • jim_thompson5910
you're given the graph
NathanJHW
  • NathanJHW
Oh duh. -2,1,3
jim_thompson5910
  • jim_thompson5910
yep
NathanJHW
  • NathanJHW
So the answer would be -2, 1 , 3?
NathanJHW
  • NathanJHW
@jim_thompson5910
jim_thompson5910
  • jim_thompson5910
for part A, yes
NathanJHW
  • NathanJHW
That was easier than I thought.
jim_thompson5910
  • jim_thompson5910
at those x values, the value of f '' is 0, so that's where the horizontal tangent on f ' will be
NathanJHW
  • NathanJHW
Now how do we do part B?
jim_thompson5910
  • jim_thompson5910
f(x) is concave up whenever f '' (x) > 0
jim_thompson5910
  • jim_thompson5910
again you're given the graph of f '' so just list the interval(s) when f '' is above the x axis to report the interval(s) when f is concave up
NathanJHW
  • NathanJHW
Well I don't think I can from the graph the exact values of f''
NathanJHW
  • NathanJHW
@jim_thompson5910
jim_thompson5910
  • jim_thompson5910
again they want the interval along the x axis
jim_thompson5910
  • jim_thompson5910
look at the graph. when is f '' (x) > 0 ? what x values?
NathanJHW
  • NathanJHW
-1 and 0
jim_thompson5910
  • jim_thompson5910
list it as an interval
jim_thompson5910
  • jim_thompson5910
from x = ??? to x = ???, the value of f '' (x) is positive
NathanJHW
  • NathanJHW
x=-2 to x=1?
jim_thompson5910
  • jim_thompson5910
good, where else?
NathanJHW
  • NathanJHW
x=1 to x=3
jim_thompson5910
  • jim_thompson5910
you sure?
NathanJHW
  • NathanJHW
Well its either that or x=3 to x = we don't know
jim_thompson5910
  • jim_thompson5910
but it gives you the bounds on which f '' is restricted
NathanJHW
  • NathanJHW
oh so x=3 to x=3.6
jim_thompson5910
  • jim_thompson5910
yep
jim_thompson5910
  • jim_thompson5910
"x = -2 to x = 1" can be written in interval notation as (-2,1) (-2,1) is NOT a point, it's an interval. yeah the notation is confusing sometimes
jim_thompson5910
  • jim_thompson5910
3 to 3.6 can be written as (3,3.6)
jim_thompson5910
  • jim_thompson5910
put the two together with a union symbol and you get \[\Large (-2,1) \cup (3,3.6)\]
jim_thompson5910
  • jim_thompson5910
that represents where f '' (x) > 0, where f is concave up
NathanJHW
  • NathanJHW
Wait, isn't concave up when the curve looks like a U not an upside down U? So doesn't that mean it is concave up between x=1 and x=3 only? Is that what the above expression means?
jim_thompson5910
  • jim_thompson5910
you are correct, but you're thinking of f having those qualities (of having a U or upside down U)
NathanJHW
  • NathanJHW
I'm confused.
jim_thompson5910
  • jim_thompson5910
you're given the graph of f '' and they want to know info about f
jim_thompson5910
  • jim_thompson5910
we don't know what f looks like but we can use f '' to figure out when f is concave up or down
NathanJHW
  • NathanJHW
Oh I see. I forgot the graph shows f''(x).
jim_thompson5910
  • jim_thompson5910
That's a common trick calc teachers use, so watch out
NathanJHW
  • NathanJHW
Alright, know how do we do part C?
jim_thompson5910
  • jim_thompson5910
the critical point x = 1.37 where is this point located on f '' ? is it in a concave up region? or concave down?
NathanJHW
  • NathanJHW
concave up
jim_thompson5910
  • jim_thompson5910
or put another way, if x = 1.37, is f '' positive or negative?
NathanJHW
  • NathanJHW
negative
jim_thompson5910
  • jim_thompson5910
since f '' is negative when x = 1.37, this means that x = 1.37 lies in a concave down region on f
NathanJHW
  • NathanJHW
so it's a maxima?
jim_thompson5910
  • jim_thompson5910
so this is what the small piece of f looks like |dw:1433375975175:dw|
jim_thompson5910
  • jim_thompson5910
oops my bad
jim_thompson5910
  • jim_thompson5910
swapped them lol
jim_thompson5910
  • jim_thompson5910
|dw:1433376035069:dw|
jim_thompson5910
  • jim_thompson5910
|dw:1433376059680:dw|
NathanJHW
  • NathanJHW
so it's the maxima?
jim_thompson5910
  • jim_thompson5910
relative max, yeah
NathanJHW
  • NathanJHW
and 0.97 is the minima for being positive
NathanJHW
  • NathanJHW
oops -0.97
NathanJHW
  • NathanJHW
actually it stays the same, it is the relative minima
jim_thompson5910
  • jim_thompson5910
if x = -0.97, then f '' (x) is positive making f concave up here so if x = -0.97, then there is a relative min on f at x = -0.97 you are correct
NathanJHW
  • NathanJHW
Alright, thank you for your help. Do you mind helping me with another problem if I do @jim_thompson5910 on the question?
jim_thompson5910
  • jim_thompson5910
sure, one more

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