Applications of Derivatives

Let f be the function given by f(x) = x³.  What are all values of c that satisfy the conclusion of the Mean Value Theorem on the closed interval [-1, 2]?  (No calculator)

• 0 only
• 1 only
• √3 only
• -1 and 1

If   $x=0.5$  is a critical point of  $f\left(x\right)$  and  $f''\left(x\right)\ =\ \cos^2x\cdot\ln x$  then  $x=0.5$  is a 

• local maximum

• local minimum

• neither

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CCSS.HSF.IF.C.7

 Which of the following is NOT a critical point of $f\left(x\right)\ =\ e^x\cdot x^2$ ?

•  $x=0$  

•  $x=1$  

•  $x=-2$  

Where is the point of inflection for the function  $f\left(x\right)\ =\ x^{3\ }+6x^2$  ?

•  $x=0$  

•  $x=-4$  

•  $x=-2$  

•  $x=4$  

On what interval(s) is the function
 $f\left(x\right)\ =\ x^{3\ }+6x^2$  
concave up?

•  $\left(-\infty,\infty\right)$  

•  $\left(-\infty,-2\right)$  

•  $\left(-2,\infty\right)$  

•  $\left(0,\infty\right)$  

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CCSS.HSF.IF.B.4

CCSS.HSF.IF.C.7

The following sign chart shows whether  $f'\left(x\right)$  is positive, negative, or zero.
There is/are ...

• a local maximum at   $x=-2$  

• a local maximum at  $x=4$  

• local maxima at  $x=-2$  and  $x=4$  

• no local extrema

How many points of inflection does a parabola have?

• 
  

  $0$ because the concavity of a parabola never changes
  

• $1$ because there is only one critical point

• $2$ because parabolas increase then decrease (or vice versa)
  

• Depends on the parabola

If $f\left(x\right)\ =\ \sqrt{1-x^2}$ ,

then  $f'\left(1\right)$  is...

•  $f\left(x\right)$  is not defined at  $x=1$  

•  $1$  

•  $\frac{2}{\sqrt{3}}$  

•  $f'\left(1\right)\$  does not exist

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CCSS.HSF.IF.A.2

 The average rate of change of $f\left(x\right)\ =\ \sqrt{x+1}$ on  $\left[3,8\right]$  is...

•  $\frac{1}{5}$  

•  $-\frac{1}{5}$  

•  $\frac{\sqrt{34}}{3}$  

•  $-\frac{\sqrt{34}}{3}$  

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CCSS.HSF.IF.B.6

Which of the following functions does not satisfy the conditions put forth in Rolle's Theorem?

• $f\left(x\right)\ =\ \left(x-2\right)^{\frac{2}{3}}\$ on $\left[3,5\right]$

• $f\left(x\right)\ =\ \sqrt{1-x^2}$ on $\left[-1,1\right]$

• $f\left(x\right)=\left(2x-3\right)^4$ on $\left[1,2\right]$

• $f\left(x\right)=e^{-x^2}$ on $\left[-4,4\right]$