mean value theorem

Question

please help ! Verify the function f(x)=x/x+2 satisfies the hypothesis of the mean value theorem on [1,4]. Then find all numbers c that satisfy the conclusion of the mean value theorem.Apparently  I am getting a different answer 

Answer 1

The Mean Value Theorem states that there is some number x=c, which is between the interval, in which its derivative at that point is equal to the slope of the line between the given interval.

 

slope=(f(4)-f(1))/(4-1)=(4/2-1/-1)/3=(2+1)/3=3/3=1

 

f'(x)=[(x-1)*1-x*1]/(x-2)^2=(x-1-x)/(x-2)^2=-1/(x-2)^2

 

f'(c)=-1/(c-2)^2=1 This implies -1=(c-2)^2=c^2-4c+4

 

c^2-4c+5=0

 

c=(4±√(4^2-4(1)(5))/2=(4±√(16-20))/2=(4±√-4)/2=(4±i2)/2

 

Hence, there is no real number that would satisfy the Mean Value Theorem between the interval. Note: to satisfy the Mean Value Theorem, the function has to be continuous between the the given interval. The function is not continuous between the interval [1,4] due to it being undefined when x=2, f(2)=2/(2-2)=2/0, Moreover, its derivatives has to be continuous between the given interval to satisfy the Mean Value Theorem.