Suppose that f is a continuous function on [0,4], f(0)=1 and 2≤f'(x)≤5. Show that 9≤f(4)≤21.?

Answer 1

since m = (change in y) / (change in x) ,
then (change in y) = (m)(change in x)
If we know where the point starts (0,1)
then the change in y cannot exceed (change in x) times (max slope)
and must be more than (change in x) times (min slope)

f(0) = 1 ===> change in x is 4, so min change in y is 4*2 = 8 ===> 8+1 =9, must be at least 9
f(0) = 1 ===> change in x is 4, so max change in y is 4*5 = 20 ===> 20+1 =21, must be at most 21

Answer 2

If f is a straight line, the result can be shown. If f'(x) is a straight line, the result cannot be shown.