Let f(x) be a function such that f(0)=2 and 1 is less then or equal to f '(x) is less then or equal to 3 for all values of x.
Find the largest and smallest possible values of f(3).
we can estimate f(3) by linear means. the minimum rate of change is 1, so the minimum value for f(3) is obtained if the rate of increase is constant at the minimum value. that is, when f(x) is a straight line with slope 1. find this line and you can find the minimum f(3). similarly, find f(x) such that it is a straight line with a slope of 3 and you can find f(3). you have the y-intercept and the slopes, finding these lines are easy