Unit 5 Assessment

Which of the following functions of x is guaranteed by the Extreme Value Theorem to have an absolute maximum on the interval [0, 4]?

• $y=\tan\ x$

• $y=\tan^{-1\ }x$

• $y=\frac{x^2-16}{x^2+x-20}$

• $y=\frac{1}{e^x-1}$

Let f be the function defined by  $f\left(x\right)=\left(x+x^2\right)e^{-2x}$  On which of the following open intervals is f increasing?

•  $\left(-\infty,\ \frac{-3-\sqrt{5}}{2}\right)\ and\ \left(\frac{-3+\sqrt{5}}{2},\ \infty\right)$  

•  $\left(-\infty,\ -1\right)\ and\ \left(0,\ \infty\right)$  

•  $\left(-\infty,\ -\frac{\sqrt{2}}{2}\right)\ and\ \left(\frac{\sqrt{2}}{2},\ \infty\right)$  

•  $\left(-\frac{\sqrt{2}}{2},\ \frac{\sqrt{2}}{2}\right)$  

Let f be the function defined by  $f\left(x\right)=\left(\sin\ x\right)e^{-x}$  on the interval  $\left[-\frac{\pi}{4},\ \frac{\pi}{2}\right]$  . On which of the following open intervals is f increasing?

•  $\left(-\frac{\pi}{4},\ \frac{\pi}{2}\right)$  

•  $\left(0,\ \frac{\pi}{2}\right)\ only\$  

•  $\left(\frac{\pi}{4},\ \frac{\pi}{2}\right)\ only$  

•  $\left(-\frac{\pi}{2},\ \frac{\pi}{4}\right)$  

Let f be a differentiable function with a domain of (0, 5). It is known that f'(x), the derivative of f(x), is negative on the intervals (0, 1) and (2, 3) and positive on the intervals (1, 2) and (3, 5). Which of the following statements is true?

• f has no relative minima and three relative maxima

• f has one relative minimum and two relative maxima

• f has two relative minima and one relative maximum

• f has three relative minima and no relative maxima

Let f be the function defined by $f\left(x\right)=12x-x^3.$  What is the absolute minimum value of f on the closed interval [0, 3]?

• -16

• 0

• 9

• 16

 Let g be the function given by $g\left(x\right)=3x^4-8x^3.$  At what value of x on the closed interval [-2, 2] does g have an absolute maximum?

• -2

• 0

• 2

•  $\frac{8}{3}$  

Let f be a twice-differentiable function. Selected values of f' and f'' are shown in the table. Which of the following statements is true?

l. f has neither a relative minimum nor a relative maximum at x = 1

ll. f has a relative maximum at x = 1

lll. f has a relative maximum at x = 4

• l only

• ll only

• lll only

• l and lll only

The function g shown in the figure above is continuous on the closed interval  $\left[0,\ x_6\right]$  and differentiable on the open interval  $\left(0,\ x_6\right)$  where  $x_1,\ x_2,\ x_3,\ x_4,\ x_5\ and\ x_6$  are points on the x-axis. Based on the graph, what are all values of x that satisfy the conclusion of the Mean Value Theorem applied to g on the closed interval  $\left[0,\ x_6\right]$  ?

•  $x_3$  only, because this is the value where g(x) equals the average rate of change of g on  $\left[0,\ x_6\right]$  

•  $g'\left(x\right)=0\ on\ \left[0,\ x_6\right]$   $x_2\ and\ x_4$  only, because these are the values where 

•  $x_1\ and\ x_5$  only, because these are the values where the instantaneous rate of change of g at those values is equal to the average rate of change of g on  $\left[0,\ x_6\right]$  

•  $x_1,\ x_3,\ and\ x_5\$  only because these are the values where either the instantaneous rate of change of g at the value is equal to the average rate of change of g on  $\left[0,\ x_6\right]$  or the value of g(x) is equal to the average rate of change of g on  $\left[0,\ x_6\right]$  

The graph of f", the second derivative of the function f, is shown above on the interval $0\le x\le6$  . Which of the following could be the graph of f?

• 
• 
• 
• 

Let f be a twice-differentiable function. Which of the following statements are individually sufficient to conclude that x=2 is that location of the absolute maximum of f on the interval [-5, 5]?
l.  $f'\left(2\right)=0$  
ll. x = 2 is the only critical pint of f on the interval [-5, 5], and  $f"\left(2\right)<0.$  
lll. x = 2 is the only critical point of f on the interval [-5, 5], and  $f\left(-5\right)<f\left(5\right)<f\left(2\right).$  

• ll only 

• lll only

• l and ll only

• ll and lll only