4. Let

a. How would you define f at 0 (i.e., f(0)) so that f(x) is continuous on the domain x >= 0? Why?

b. Classify the critical points of f(x) on the domain x > 0.

Solution a

By definition, function f(x) will be continuous at 0 if and only if

f is defined at 0 so that

To evaluate

replace x by 1/u and apply L'Hospitals' rule. One gets

So, define f at 0 by f(0) = 0. Then, by the above, f(x) is continuous at x = 0.

In addition f(x) is always greater than or equal to 0, is 0 at x = 0,is 0 at x = 1 and goes to infinity as x goes to infinity. Thus it must have a local maximum when ln(x) = -2/3 (i.e., when x = e^-2/3) and a global minimum (of 0) at x = 1.