Does the function satisfy the hypotheses of the Mean Value Theorem on the given

Question

Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? f(x) = (x/x + 7) , [1, 14] Yes, f is continuous on [1, 14] and differentiable on (1, 14). No, f is continuous on [1, 14] but not differentiable on (1, 14). Yes, it does not matter if f is continuous or differentiable; every function satisfies the Mean Value Theorem. There is not enough information to verify if this function satisfies the Mean Value Theorem. No, f is not continuous on [1, 14]. If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem. (Enter your answers as a comma-separated list. If it does not satisfy the hypotheses, enter DNE). c =

Explanation / Answer

f(x) = (x/(x + 7))

f(x) is not defined when x+7=0==>x=-7

but x=-7 sis not in given interval. so function is continous in [1,14]

f '(x) = (1(x + 7)-(1+0)x)/(x+7)2

f '(x) = (7)/(x+7)2

f '(c) = (7)/(c+7)2

f '(c) =[f(14) -f (1) ]/(14-1)

(7)/(c+7)2=[ (14/(14 + 7)) - (1/(1 + 7)) ]/(14-1)

(7)/(c+7)2=[ (2/3) - (1/8) ]/(13)

(7)/(c+7)2=[ (13)/(24) ]/(13)

(c+7)2=7*24

(c+7)=12.96

c=5.96

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