Section 4.1Using First and Second Derivatives

• Let’s see what we remember about derivatives of a function and its graph– If f’ > 0 on an interval than f is

• Increasing

– If f’ < 0 on an interval than f is• Decreasing

– If f’’ > 0 on an interval than the graph of f is• Concave up

– If f’’ < 0 on an interval than the graph of f is• Concave down

• Consider the function

• Let’s find where it is increasing, decreasing, concave up, and concave down algebraically and then check that with the graph

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Critical Points

• For any function f, a point p in the domain of f where f’(p) = 0 or f’(p) is undefined is called a critical point of the function – The critical value of f is the function value, f(p)

where p is the critical point– Critical points are used to determine relative

extrema

Relative Extrema

• f has a local maximum at x = p if f(p) is equal to or larger than all other f values near p– If p is a critical point and f’ changes from positive to

negative at p, then f has a local maximum at p

• f has a local minimum at x = p if f(p) is equal to or smaller than all other f values near p– If p is a critical point and f’ changes from negative to

positive at p, then f has a local maximum at p

• Since in the previous cases we were using the first derivative, we were using the first derivative test to check for relative extrema

The Second Derivative Test for Relative Extrema

Suppose f’(p) = 0 and thus p is a critical point of f

• If f’’(p) < 0 then– f has a local maximum at p

• If f’’(p) > 0 then– f has a local minimum at p

• If f’’(p) = 0 then– The test tells us nothing

• Places where the graph switches concavity are called inflection points

• How can we identify inflection points?– Where the second derivative is zero or undefined

Example

• For the following function find where it is increasing, decreasing, concave up, and concave down algebraically and identify any relative extrema and/or inflection points

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