Finding extrema

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

If   $x=e$  is a critical point of  $f\left(x\right)$  and  $f''\left(x\right)\ =\ \ln\left(x\right)-1$  then  $x=e$  is a 

• local maximum

• local minimum

• neither

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

•  $x=0$  

•  $x=1$  

•  $x=-2$  

Which of the following are NOT critical points of $f\left(x\right)\ =3^{4^x}$ ?

•  $x=0$  

•  $x=-7$  

•  $x=-\frac{\ln3}{\ln7}$  

•  $x=1$  

Where does the global maximum of $f\left(x\right)\ =\ e^{-x^2}$  occur on  $\left[-1,2\right]$ ? 

•  $x=-1$  

•  $x=0$  

•  $x=1$  

•  $x=2$  

Where does the global minimum of  $f\left(x\right)\ =\ x^3-6x^2$  occur on the interval $\left[-2,6.5\right]$ ?

•  $x=-2$  

•  $x=0$  

•  $x=4$  

•  $x=6.5$  

Over what interval(s) is f(x) increasing? (Be careful this is a graph of f'!)

• (-∞, -3) ∪ (1, ∞)
• (-3, 1)
• (-5, 0) ∪ (2, ∞)
• (-5, ∞)

Over what interval(s) is f(x) decreasing?

• (-3, 1)
• (-∞, -5) ∪ (0, 2)
• (-∞, -3) ∪ (1, ∞)
• (-5, 0) ∪ (2, ∞)

If (a,b) is a local maximum, then what will be true about f''(a)?

• It's positive
• It's negative
• It's zero
• Cannot be determined

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

• It's postive
• It's negative
• It's zero
• Cannot be determined