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:

attachment?

step-by-step explanation:

x = -3

y = -18

step-by-step explanation:

to solve by substitution, substitute one variable and solve for the other. then substitute again to find the remaining variable.

x-1/3(3x-9) = -3 since y=3x-9 substittute y.

x-1/3(3x-9)=-3

x + x +3 = -3

2x+3=-3

2x+3-3= -3-3

2x=-6

x=-3

substitute again into y=3x-9

y=3(-3)-9

y=-9-9

y=-18