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Curve Sketching: Role of first and second derivatives

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- Increasing/decreasing functions and the sign of the derivative
- Concavity and the sign of the second derivative

- If f is defined on an interval then
- f is increasing if f(x1) < f(x2) whenever x1<x2
- f is decreasing if f(x1) > f(x2) whenever x1<x2
- f is constant if f(x1) = f(x2) for all points x1 and x2

- Fact: If f is continuous on [a,b] and differentiable in (a,b), then
- f is increasing on [a,b] if f (x)>0 for all x in (a,b)
- f is decreasing on [a,b] if f (x)<0 for all x in (a,b)
- f is constant on [a,b] if f (x)=0 for all x in (a,b)

- Find the intervals on which the following functions are increasing or decreasing

- Concavity measures the curvature of a function. Concave up=“holds water”. Concave down=“spills water”.
- If f is differentiable on an open interval I then f is concave up if f is increasing there, and f is concave down if f is decreasing there
- Conclusion: f concave up on I if f >0 on I and f concave down on I if f <0 on I

- Find the intervals on which the following functions are concave up and concave down

- If f changes concavity at x0 then f has an inflection point at x0 (i.e. f changes sign at x0)
- Examples: Find the inflection points of the following functions

- Find the inflection points of the following functions and sketch their graphs