anonymous
  • anonymous
Suppose that we know that f(x) is a continuous and differentiable function on [2,14]. Also, suppose we know that f(2)=9 and f'(x) is greater than or equal to -12. What is the smallest possible value of f(14)?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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amistre64
  • amistre64
would this be a straight line from 2,14 running at a slope of 12?
anonymous
  • anonymous
(I'm supposed to use the mean value theorem, or at least that's what it says on the worksheet.)
anonymous
  • anonymous
(I'm supposed to use the mean value theorem, or at least that's what it says on the worksheet.)

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More answers

anonymous
  • anonymous
@amistre64 - I have no idea! I'm complete lost on this one. Don't even know where to start.
amistre64
  • amistre64
f'(2) =>12....6x? maybe
anonymous
  • anonymous
Well, if f'(x) = -12 throughout the interval then it would be a straight line from (2,9) with a slope of -12. So that would be the minimum possible value given what we know.
anonymous
  • anonymous
Sorry, @polpak - how do we know that?
amistre64
  • amistre64
ahh,....(-)12 ... that was hiding in the corner
anonymous
  • anonymous
We are given that f'(x) is greater than or equal to -12.
anonymous
  • anonymous
So if it maintained the smallest possible derivative over the whole interval we would have a line with slope of -12.
myininaya
  • myininaya
i got the smallest possible value is -135 for f(14)
amistre64
  • amistre64
thats what I thought :)
amistre64
  • amistre64
but with a 12 lol
anonymous
  • anonymous
9-12*12 = 9-144 = -135
myininaya
  • myininaya
yay thats what i got except differently
myininaya
  • myininaya
-12<=[f(2)-f(14)]/[2-14]
anonymous
  • anonymous
Does that make sense Chap?
anonymous
  • anonymous
How do you figure what f(14) is?
myininaya
  • myininaya
chap you can also solve the above for f(14) resulting in f(14)>=(-135)
myininaya
  • myininaya
I used the mean value thm just like you said
anonymous
  • anonymous
Right. But I have -12<= 9 - f(14)
anonymous
  • anonymous
Oops wait
anonymous
  • anonymous
I have -12 <= 9 - f(14) ----------- -12
myininaya
  • myininaya
multipliy -12 on both sides don't forget to flip the inequality when you multipliy or divide by a negative
anonymous
  • anonymous
Sorry )= I don't understand how to do that.
anonymous
  • anonymous
\[-12 \le \frac{9- f(14)}{-12} \implies 144 \ge 9-f(14)\]
myininaya
  • myininaya
how do you solve x/5=4
anonymous
  • anonymous
\[-4a < b \implies a \gt \frac{b}{-4}\]
anonymous
  • anonymous
Okay. But how does 135 >= - f(14) solve for f(14)?
anonymous
  • anonymous
Multiply by -1
myininaya
  • myininaya
now multipliy both sides by negative 1 don't forget to flip
anonymous
  • anonymous
135 <= f(14)
myininaya
  • myininaya
right!
myininaya
  • myininaya
wait where's the negative?
anonymous
  • anonymous
That's the answer?
anonymous
  • anonymous
-135 <= f(14)
myininaya
  • myininaya
ok so we have f(14)>=-135 this means the smallest that f(14) can be is -135
anonymous
  • anonymous
OH! I thought f(14) would be positive. Thank you so much!
anonymous
  • anonymous
It might be positive!
anonymous
  • anonymous
We only know that the smallest it can be is -135. It could be very very large.
myininaya
  • myininaya
it could be 1111111111111111111111
anonymous
  • anonymous
over 9000!
myininaya
  • myininaya
5 trillion billion if that means anything

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