Prove that 9=f(4)=21 using information given

Question

Suppose that f(0)=1 and 2≤f'(x)≤5 for all x in interval (0,4). Show that 9≤f(4)≤21

Answer 1

Dear Jo,

 

Here are some hints to help you along.

 

The proof is to show that the lowest rate of increase will give f(4) = 9 and the highest rate of increase will give f(4) = 21. Therefore 9 =< f(4) =< 21.

 

The lower rate of increase is assuming f'(x) = 2 in the interval (0,4). The upper rate of increase is assuming f'(x) = 5 in the interval (0,4).

 

If f'(x) is constant within the interval, then f(x) is a straight line. Use this to establish the lower and upper limit.

 

Good luck