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**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 ____.)

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**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

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**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: _______

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In this section, we will investigate some graphical relationships between a function and its second derivative.

In this section, we will investigate some graphical relationships between a function and its second derivative.

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