Unit 5 Review

When  $f'\left(x\right)$  changes from negative to positive, there is...

• a maximum.

• a minimum.

• no extrema.

  If  $\left(a,b\right)$ is a local minimum, then what will be true about  $f'\left(a\right)$  ? 

• It's positive

• It's negative

• It's zero

• Cannot be determined

The graph of $f\left(x\right)=2x^3+x^2+20x+1$   has a relative minimum at

•  $x=-2$  

•  $x=\frac{5}{3}$  

•  $x=2$  

•  $x=1$  

Find all the critical numbers for the function:

 $g\left(x\right)=x\sqrt{2x+1}$  

•  $-\frac{1}{3},\ -\frac{1}{2}$  

•  $0,\ \frac{1}{3},\ \frac{1}{2}$  

•  $\frac{1}{3},\ -\frac{1}{2}$  

•  $\frac{1}{3},\ \frac{1}{2}$  

Consider the function $f\left(x\right)=x^3+3x^2-45x+5$  , then 

•  $f$  is decreasing on  $\left(-5,3\right)$  and increasing on  $\left(-\infty,-5\right)\cup\left(3,\infty\right)$  

•  $f$  has a relative minimum at  $x=3$  

•  $f$  has a relative maximum at  $x=-5$  

• All of the above 

• None of the above 

Find the positive value $x=c$ 

, that satisfies the conclusion the Mean Value Theorem for Derivatives given  $f\left(x\right)=3x^2-5x+4$  on the interval   $\left[2,5\right]$   

•  $\frac{11}{6}$  

•  $\frac{13}{6}$  

•  $\frac{7}{2}$  

•  $\frac{23}{6}$  

Find all the open intervals where $f\left(x\right)=\frac{x^2}{x^2+4}$ is decreasing. 

•  $\left(-\infty,0\right)$  

•  $\left(-\infty,\infty\right)$  

•  $\left(-2,2\right)$  

•  $\left(0,\infty\right)$  

Which of the following statements is true for $g\left(x\right)=x^{3\ }+6x^2+9x+2$  

•  $g$  is increasing on  $\left(-3,-1\right)$  

•  $g$  is increasing on  $\left(-2,\infty\right)$  

•  $g$  is decreasing on  $\left(-\infty,-3\right)$  

•  $g$  is decreasing on  $\left(-3,-1\right)$  

Rolle's Theorem does not apply to the function $f\left(x\right)=\frac{2}{\left(x+1\right)^2}$ on the interval  $\left[-2,0\right]$ . Explain why. 

•  $f$  is not continuous on  $\left[-2,0\right]$  

•  $f$  is not differentiable at  $x=-1$  

•   $f$  is not continuous on  $\left[-2,0\right]$  and not differentiable at  $x=-1$  

•  $f\left(-2\right)\ne f\left(0\right)$  

Find the values of $x$ 

that give relative extrema for the function  $h\left(x\right)=3x^5-5x^3$  

• Relative maximum:  $x=-1$  ; Relative minimum:  $x=1$   
  

• Relative maximum:  $x=0$ ; Relative minimum:  $x=\sqrt{\frac{3}{5}}$ 

• Relative maximum:  $x=\pm1$  ; Relative minimum:  $x=0$  

• Relative maximum:  $x=0$  ; Relative minimum:  $x=\pm1$