Section 4.6

The Shape of a Graph II

The first derivative tells us if the graph is going up or down. The **second derivative** tells us how the graph bends. Is it cupped like a bowl or a frown?

1

Concavity & The Second Derivative

Concave Up

"Holds Water." (Tangent lines sit below graph)

Concave Down

"Spills Water." (Tangent lines sit above graph)

Inflection Point: A point where concavity changes sign (from Up to Down or vice versa). Occurs where or is undefined (and changes sign).

2

Worked Example

Analyzing Concavity of xe^x

Find the intervals of concavity and inflection points for .

The Tangent Sled

Concave Down

Tangent is ABOVE the curve (Graph is 'Sad')

Drag to slide the tangent line along the curve.

1. Second Derivative
2. Check Sign of f''

is always positive. Only matters.

  • : Neg (Down)
  • : Pos (Up)
3. Conclusion

Since concavity changes at , there is an Inflection Point at .

3

Level Up Examples

Example A: Polynomial Flatness

Find inflection points of .

1. Derivatives:

2. Candidates:
.
3. Test Signs:
  • : UP
  • : DOWN
  • : UP

Inflection points at 0 and 2.

Example B: Trig Concavity

Describe concavity of on .

1. Derivatives:

2. Set to Zero:
.
.
3. Test Concavity:
  • (e.g. 0): DOWN
  • (e.g. ): UP
  • (e.g. ): DOWN
5

Practice Quiz

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