Let f be the function defined by f(x) = x + ln x. what is the value of c for which the instantaneous rate of change of f at x = c is the same as the average rate of change of f over [1,4]?

The value will be "C = 2.164" for which the instantaneous rate of change is similar to the average rate.

Given:

Instantaneous rate of x=c is,

at x=c

              at x=c

              ...(2)

Now,

The average rate of change over [1, 4] will be:

→              

                  ...(3)

Since both are equal,

From equation "2" and "3", we get

→    

→

→      

→        

Thus the above answer is right.

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is differentiable for all , so by the mean value theorem there is some for which

We have derivative

so that satisfies

which can be rewritten using the change of base formula, if you wish: