Math 101 Section 4.1: First and second derivative tests Date:

Our big goal for today is to think about the graphs of functions and how to use derivatives to learn about where a function is locally maximized or minimized and where a function changes concavity.

Example. Let’s consider the function f(x) = x3 − 9x2 − 48x + 52 and we’ll discuss the new terms local minimum and local maximum in the context of this example. 400

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• The point x = p is a local minimum if f(p) ≤ f(x) for x near p.

• The point x = p is a local maximum if f(p) ≥ f(x) for x near p.

• The point x = p is a critical point if f 0(p) = 0 or f 0(p) is undefined.

Example. Find f 0(x) and the critical points in our previous example. Then sketch a number line showing where f 0(x) is positive, negative, and 0. The First Derivative Test. Suppose that x = p is a critical point of f.

• If f 0 changes from negative to positive at p, then f has a local minimum at p.

• If f 0 changes from positive to negative at p, then f has a local maximum at p.

1 4 1 3 2 Example. Use the first derivative test to classify the local extrema of f(x) = 4 x + 3 x − x .

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Example. Use the first derivative test to classify the local extrema of f(x) = 3x4 − 4x3 − 12x2 + 3. Example. Go back to the sketch of f(x) = x3 − 9x2 − 48x + 52 from the first page. Using your sketch, what you can you say about the concavity of the graph near each critical point? Confirm your answer by computing f 00(p) for each critical point p.

The Second Derivative Test. Suppose that x = p is a critical point of f.

• If f 00(p) > 0, then f has a local minimum at p.

• If f 00(p) < 0, then f has a local maximum at p.

• If f 00(p) = 0, then the test is inconclusive (we use the first derivative test instead).

1 4 1 3 2 Example. Use the second derivative test to classify the local extrema of f(x) = 4 x + 3 x − x . Example. Use the second derivative test to classify the local extrema of f(x) = 3x4 − 4x3 − 12x2 + 3.

Questions. Think about and answer the following questions.

1 4 3 2 1. Try the first derivative test on f(x) = 4 x − 4x + 18x . What can you conclude?

2. If f 0(p) = 0, does this guarantee that x = p is a local maximum or minimum?

3. Does f(x) = |x| have critical points? What does the first derivative test say about f(x) = |x|?