56. Limit Involving Radicals and as x Approaching ∞
We have done limits at infinity (i.e., limx∞f(x) and limx–∞f(x) where f(x) is a rational function), what happen if f(x) is not a rational function? For example,
(Usually you don’t see this limit because
________________, but if you do see it, you will say the limit is ____.)

57. First and Second Derivative Tests for Local Maximum and Minimum
Recall that if f (c) = 0, most likely f has a local extremum at x = c. However, it’s not a must (recall the example f(x) = x3, where f (0) = 0, but f doesn’t have a local extremum at x = 0—but rather, it is an inflection point). Even if f has a local extremum at x = c when f (c) = 0, how can we tell it is a local maximum or minimum? We need one of the following two tests, one of which we have already learned.
The First Derivative Test
Suppose that c is a critical number of a continuous function f (i.e, f (c) = 0).
i. If f  changes from positive to negative at c, then f has a local maximum at c.
ii. If f  changes from negative to positive at c, then f has a local ________ at c.
iii. If f  does not change sign at c (for example, if f  is positive on both sides of c or negative on both sides,
then f has no local maximum or minimum at c.
Examples:
The other test is called the Second Derivative Test, which is a consequence of the Concavity Test.
The Second Derivative Test
Suppose f (c) exist.
i. If f (c) = 0 and f (c) > 0, then f has a local ___________ at c.
ii. If f (c) = 0 and f (c) < 0, then f has a local ___________ at c.
iii. If f (c) = 0 and f (c) = 0, then the test is _____________.
f(x) = x3

58. Graphs of f, f  and f  Given the graph of function f on the right:
1. Indicate the intervals for which f is increasing: ______________
2. Indicate the intervals for which f is decreasing: ______________
3. Indicate the x-values for which f has a local maximum: _______
4. Indicate the x-values for which f has a local minimum: ________
5. Indicate the intervals for which f is concave up: _____________
6. Indicate the intervals for which f is concave down: ___________
7. Indicate the x-values for which f has an inflection point:_______
Given the graph of function f on the right:
1. Indicate the intervals for which f is increasing: ______________
2. Indicate the intervals for which f is decreasing: ______________
3. Indicate the x-values for which f has a local maximum: _______
4. Indicate the x-values for which f has a local minimum: _______
Given the graph of function f on the right:
1. Indicate the intervals for which f is concave up: ______________
2. Indicate the intervals for which f is concave down: ______________
3. Indicate the x-values for which f has an inflection point: _______