Extremas, Inflections, and Graphs

The graph of a twice-differentiable function f is shown. Which of the following is true?

• $f\left(-2\right)<f'\left(-2\right)<f''\left(-2\right)$

• $f\left(-2\right)<f''\left(-2\right)<f'\left(-2\right)$

• $f''\left(-2\right)<f'\left(-2\right)<f\left(-2\right)$

• $f'\left(-2\right)<f\left(-2\right)<f''\left(-2\right)$

The graph of $f'\left(x\right)$  is shown.  Which of the following statements is not true?

•  $f$  is increasing on [2, 3]

•  $f$  has a local minimum at  $x=1$  

•  $f$  has a local maximum at  $x=0$  

•  $f$  is differentiable at  $x=3$  

If $y$ is a function of  $x$  such that  $y'>0$  and  $y''<0,$  which could be  $y=f\left(x\right)?$  

• A

• B

• C

• D

 $f$  is continuous,  $f'\left(0\right)$  does not exist,  $f'\left(2\right)=0$  and  $f''\left(x\right)<0.$  Which could be the graph of  $f?$  

• A

• B

• C

• D

If $f\left(3\right)=7,\ f'\left(3\right)=0,$  and  $f''\left(3\right)=4,$  then which of the following statements must be true?

•  $f$  attains a maximum at  $x=3$  

•  $f$  attains a minimum at  $x=3$  

•  $f$  has a point of inflection at  $x=3$  

• Nothing can be determined given the information

The graph of the derivative of $f$ is shown.  Which of the following could be the graph of  $f?$ 

• A

• B

• C

• D

If $f'\left(x\right)=\left(2x-1\right)^2\left(3x+1\right)^3\left(x-1\right),$ then  $f$ has which of the following relative extrema? 

• I only

• I and II only

• III only 
  

• I and III only

The graph of $y=5x^4-x^5$ 

has a point of inflection at 

• (0, 0) only

• (4, 256) only

• (3, 162) only

• (0, 0) and (3, 162)

At $x=0,$ which of the following is true of the function  $f$ defined by  $f\left(x\right)=x^2+e^{-2x}?$  

•  $f$ has a relative minimum

•  $f$ has a relative maximum

•  $f$ is increasing

•  $f$ is decreasing

The graph of $y=\frac{-5}{x-2}$ is concave downward for all values of  $x$ such that 

•  $x>2$  

•  $x<2$  

•  $x>0$  

•  $x<0$  

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