136.152: Test #5

20 minutes

Name:_____________________ Student Number: _______

1. Differentiate the following functions. Do NOT simplify.

(a)

(b)

(a)

(b) .

2.  Find and classify all critical points of the function .

and the only critical number comes from solving , yielding . Since is positive, we get a local minimum there. (Alternatively, analyze the intervals where the functions increases/decreases to get to the same conclusion).

3. Find the inflection points and the intervals where the function is concave up, where it is concave down.

, and the only possible inflection point is (obtained by solving ). That point splits the domain into the intervals and . Chose a number in each of these two intervals and see the sign of the second derivative: (negative), so the function is concave down over ; (positive), and so the function is concave up over . So, we do have an inflection point at .