Concavity and Points of Inflection

5-6 Concavity

Guided Notes

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Multiple Choice

The concavity of a function is described by its _______________.

• first derivative
• second derivative
• third derivative
• expression

Multiple Select

Discuss the concavity for the function

$$f\left(x\right)=2x^3+6x^2-4$$  Select ALL the correct answers

• Concave up in   $$\left(-\infty,-1\right)$$   

• Concave down in    $$\left(-\infty,-1\right)$$  

• Concave down in  $$\left(-1,\infty\right)$$  

• Concave down in     $$\left(-2,0\right)$$  

• Concave up in    $$\left(-1,\infty\right)$$  

Multiple Choice

Discuss the concavity and the point(s) of inflection (if any) of the function:

$$f\left(x\right)=\frac{2}{\left(x-1\right)^3}$$  

• Concave down in  $$\left(-\infty,1\right)$$  , Concave up in $$\left(5,\infty\right)$$  and the point of inflection at x=1.
  

• Concave up in  $\left(-\infty,1\right)$  , Concave down in $$\left(1,\infty\right)$$  and the point of inflection at x=1.
  

• Concave down in  $\left(-\infty,1\right)$ and $$\left(2,\infty\right)$$ , with no point of inflection.
  

• Concave up in  $\left(-\infty,1\right)$ and $$\left(2,\infty\right)$$ , with no point of inflection.

• Concave up in $$\left(-\infty,\infty\right)$$  with no point of inflection.

Multiple Select

Find the point(s) of inflection (if it exists) of the graphed function.

• No point of inflection

• Points 3 and 5

• Points 1, 3, 5 and 7

• Points 1, 2, 3, 4, 5, 6 and 7

• Points 2, 4 and 6

Multiple Choice

Find the point of inflection (if it exists) of the function:

$$f\left(x\right)=x^3-12x^2+7$$  
Select ALL the correct answer(s).

• Where x = 8

• Where x = 4

• Where x = 0

• No inflection point

• Where x = -8