suppose f is a differentiable function on the open interval

suppose f is a differentiable function on the open interval (1,4) and continuous of [1,4].

can f(4) possibly be?

Solution

By the mean value theorem,there have to be a c such that

f\'(c)=f(4)-f(1)/(4-1)

f\'(c)=f(4)-10/3

and f\'(x)>=5

Therefore f(4)-10/3>=5

f(4)>=25

Therefore the least value of f(4) is 25