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anonymous
 one year ago
Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval?
f(x) = x/x+6
[1, 12]
If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem. (Enter your answers as a commaseparated list. If it does not satisfy the hypotheses, enter DNE). Ive tried to do it multiple ways but i cant get the right answer. Please help!
anonymous
 one year ago
Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? f(x) = x/x+6 [1, 12] If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem. (Enter your answers as a commaseparated list. If it does not satisfy the hypotheses, enter DNE). Ive tried to do it multiple ways but i cant get the right answer. Please help!

This Question is Closed

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4the function is \[\Large f(x) = \frac{x}{x+6}\] right?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4which x value makes the denominator zero?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4is 6 in the interval [1,12] ?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4so the function is continuous on [1,12] that allows us to use the MVT properly here

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4if there was a discontinuity on [1,12] then we couldn't use the MVT

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay, I understand that part, the equation part is whats given me a lot of trouble, i dont have examples like this one

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4Mean Value Theorem If f(x) is continuous on [a,b] and differentiable on (a,b), then there exists at least one value of c such that \[\Large f \ '(c) = \frac{f(b)  f(a)}{ba}\]

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4what you have to do is find the secant slope through (1, f(1)) and (12,f(12)) then find f ' (x) set f ' (x) equal to the secant slope and solve for x

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so i would substitute 1 for a and 12 for b? that gives me 1?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4you said `that gives me 1?` what do you mean exactly? that's the secant slope you got?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yeah if i substitute the a for 1 and 12 for b, that equals to 11/11 so 1? i dont think im doing this right

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4that's not the correct secant slope

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4\[\Large f(x) = \frac{x}{x+6}\] \[\Large f(1) = \frac{1}{1+6}\] \[\Large f(1) = \frac{1}{7}\]

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4\[\Large f(x) = \frac{x}{x+6}\] \[\Large f(12) = \frac{12}{12+6}\] \[\Large f(12) = \frac{12}{18}\] \[\Large f(12) = \frac{2}{3}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0and then i would do the same for f(12)?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay I got that part!

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4you should have this next \[\Large f \ '(c) = \frac{f(b)  f(a)}{ba}\] \[\Large f \ '(c) = \frac{f(12)  f(1)}{121}\] \[\Large f \ '(c) = \frac{\frac{2}{3}  \frac{1}{7}}{121}\] \[\Large f \ '(c) = \frac{\frac{11}{21}}{11}\] \[\Large f \ '(c) = \frac{1}{21}\]

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4now you need f ' (x)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so then i would substitute 1/21 for x in this 6/(6+x)^2 right?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4you set them equal to one another and solve for x

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I set what equal to each other?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.41/21 and that f ' (x) you got

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i end up with 6=1/21x^2+4/7x+12/7

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0what would i do w the other x?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4\[\Large f \ '(x) = \frac{6}{(x+6)^2}\] \[\Large \frac{1}{21} = \frac{6}{(x+6)^2}\] \[\Large 1*(x+6)^2 = 21*6\] \[\Large (x+6)^2 = 126\] keep going to solve for x

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0got 6+3root14 and 63root 14?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4those are the correct solutions to the equation you now need to check if they lie in the interval [1,12]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay so it would only be 6+3root14

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4yes since \(\large 6+3\sqrt{14} \approx 5.22\) which is in the interval [1,12]

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.4the other value is 17.22 which is not in that interval
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