Concavity Warm-Up

To determine concavity of a function, you find the:

• 1st Derivative

• 2nd Derivative

• x-intercepts

• y-intercept

Points of Inflection occur ONLY when the function changes from CONCAVE UP to CONCAVE DOWN.

• TRUE

• FALSE

Tags

CCSS.HSF.IF.B.4

CCSS.HSF.IF.C.7

Determine the concavity of the function:

 $f\left(x\right)=x^4-6x^3-60x^2+5x-12$  

STEP ONE:  Find the derivative:

•  $f'\left(x\right)=4x^3-18x^2-120x+5$  

•  $f'\left(x\right)=12x^2-36x-120$  

•  $f'\left(x\right)=24x-36$  

•  $f'\left(x\right)=24$  

Determine the concavity of the function:

 $f\left(x\right)=x^4-6x^3-60x^2+5x-12$  

STEP TWO:  Find the second derivative:

•  $f''\left(x\right)=4x^3-18x^2-120x+5$  

•  $f''\left(x\right)=12x^2-36x-120$  

•  $f''\left(x\right)=24x-36$  

•  $f''\left(x\right)=24$  

Determine the concavity of the function:

 $f\left(x\right)=x^4-6x^3-60x^2+5x-12$  

STEP THREE:  Set second derivative equal to zero and solve...

•  $x=-2,5$  

•  $x=-5,2$  

•  $x=5$  

•  $x=-2$  

Tags

CCSS.HSA.APR.B.3

Determine the concavity of the function:

 $f\left(x\right)=x^4-6x^3-60x^2+5x-12$  

STEP FOUR:  Determine when the function is concave UP (interval notation)

•  $\left(-\infty,-2\right)\ and\ \left(5,\infty\right)$  

•  $\left(-\infty,-2\right)$  

•  $\left(-2,5\right)$  

•  $\left(5,\infty\right)$  

Determine the concavity of the function:

 $f\left(x\right)=x^4-6x^3-60x^2+5x-12$  

STEP FIVE:  Determine when the function is concave DOWN (interval notation)

•  $\left(-\infty,-2\right)\ and\ \left(5,\infty\right)$  

•  $\left(-\infty,-2\right)$  

•  $\left(-2,5\right)$  

•  $\left(5,\infty\right)$  

Determine when the function is concave UP:
 $g\left(x\right)=-x^4-4x^3-2x^2+4x$ 
NOTE:  You will need to use the Quadratic Formula.

•  $\left(\frac{-6-\sqrt{24}}{6},\frac{-6+\sqrt{24}}{6}\right)$  

•  $\left(-\infty,\frac{-6-\sqrt{24}}{6}\right)$  

•  $\left(\frac{-6+\sqrt{24}}{6},\infty\right)$  

•  $\left(-\infty,\frac{-6-\sqrt{24}}{6}\right)\ and\ \left(\frac{-6+\sqrt{24}}{6},\infty\right)$  

Tags

CCSS.HSA.REI.B.4

Determine when the function is concave DOWN:
 $g\left(x\right)=-x^4-4x^3-2x^2+4x$ 
NOTE:  You will need to use the Quadratic Formula.

•  $\left(\frac{-6-\sqrt{24}}{6},\frac{-6+\sqrt{24}}{6}\right)$  

•  $\left(-\infty,\frac{-6-\sqrt{24}}{6}\right)$  

•  $\left(\frac{-6+\sqrt{24}}{6},\infty\right)$  

•  $\left(-\infty,\frac{-6-\sqrt{24}}{6}\right)\ and\ \left(\frac{-6+\sqrt{24}}{6},\infty\right)$  

Tags

CCSS.HSA.REI.B.4