Quiz 6

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Total points: 8

Let $f(x) =e^{- \frac{5 x^{3}}{3} - 10 x^{2} - 20 x - 8} $.
1. Calculate $f'(x)$.
  
  $f'(x) = $
2. Find the critical points of $f$.
  
  Critical points =
  (If there are multiple answers, separate by commas; do not round.)
3. The critical points divide the number line into two intervals. What are these two intervals? Enter them from left to right. Enter oo for $\infty$.
  
  Interval 1 =
  
  Interval 2 =
4. On each of these intervals, $f'(x)$ does not change sign. Pick an auxiliary point in each interval, test the sign of $f'(x)$ at that point, and conclude whether $f$ is increasing or decreasing on that interval. Enter either increasing or decreasing in the answer blanks.
  
  On interval 1, $f$ is
  
  On interval 2, $f$ is
Let $h(t) = 3(t-n)^2 -6$, where $n$ is a constant parameter.
1. Find the critical points of $h$.
  
  Critical points: $t=$
  (If multiple answers, separate by commas; if no answers, enter none.)
2. Find the local extrema of $h$. (Extrema just means either a maximum or a minimum.) For each extremum calculate three things: the location of the extremum (i.e., value of $t$), the value of the extremum (i.e., value of $h(t)$), and whether it is a local maximum or a local minimum.
  
  Locations of the local extrema:
  (If multiple answers, separate by commas; if no answers, enter none.)
  
  Values of the local extrema:
  (If multiple answers, enter in the same order as above, separated by commas; if no answers, enter none.)
  
  For each extremum, enter either maximum or minimum to indicate if the extremum is a local maximum or local minimum, respectively. (If multiple answers, enter in the same order as above, separated by commas; if no answers, enter none.)
  Characterization of extrema=
  
  Need help?Hide help
  
  Hint
  For example, imagine that there were local extrema at $x=-4$, $x=-1$, and $x=3$. Imagine moreover, that $f(-4)=0$, $f(-1)=-3$, and $f(3)=4$, and that these extrema were a local maximum, a local minimum, and a local maximum, respectively. Then, the correct answer would be to enter -4,-1, 3 in the locations blank, enter 0, -3, 4 in the values blank, and maximum, minimum, maximum in the characterization blank.
  Hide help
3. Assume $n >0$. Find the global maximum and minimum of $h$ on the interval $0 \le t \le 3n$. Also indicate the location (the value of $t$) of the global maximum and global minimum.
  
  Global maximum:
  
  Location of global maximum: $t =$
  
  Global minimum:
  
  Location of global minimum: $t= $
Imagine that, if every year, a fraction $s$ of the fish are harvested from a lake, then the average number of fish in the lake will be $p(s)=9700(1-s)^4$. This means that the annual fish harvest is $h(s)= s \cdot p(s) = 9700 s (1-s)^4$. The goal is to determine the fraction of fish to harvest in order to maximize the harvest.

What is the relevant range of fraction $s$?
$ \le s \le $

What is the derivative of $h$? $\diff{h}{ s } = $

In order to maximize the fish harvest, what fraction $s$ of fish should be harvested each year? $s = $

In this case, what will be the average number of fish in the lake?
(If rounding, keep at least four digits.)

How many fish will be harvested each year, on average?
(If rounding, keep at least four digits.)
Let $f(x) =e^{\frac{4 x^{3}}{3} - 16 x^{2} + 64 x - 8} $.
1. Calculate $f'(x)$.
  
  $f'(x) = $
2. Find the critical points of $f$.
  
  Critical points =
  (If there are multiple answers, separate by commas; do not round.)
3. Find the local extrema of $f$. (Extrema just means either a maximum or a minimum.) For each extremum calculate three things: the location of the extremum (i.e., value of $x$), the value of the extremum (i.e., value of $f(x)$), and whether it is a local maximum or a local minimum.
  
  Locations of the local extrema:
  (If multiple answers, enter in increasing order, separated by commas; if no answers, enter none.)
  
  Values of the local extrema:
  (If multiple answers, enter in the same order as above, separated by commas; if no answers, enter none. If rounding, keep at least four digits.)
  
  For each extremum, enter either maximum or minimum to indicate if the extremum is a local maximum or local minimum, respectively. (If multiple answers, enter in the same order as above, separated by commas; if no answers, enter none.)
  Characterization of extrema=
  
  Need help?Hide help
  
  Hint
  For example, imagine that there were local extrema at $x=-4$, $x=-1$, and $x=3$. Imagine moreover, that $f(-4)=0$, $f(-1)=-3$, and $f(3)=4$, and that these extrema were a local maximum, a local minimum, and a local maximum, respectively. Then, the correct answer would be to enter -4,-1, 3 in the locations blank, enter 0, -3, 4 in the values blank, and maximum, minimum, maximum in the characterization blank.
  Hide help
4. Find the global maximum and global minimum of the function $f(x)$ on the interval $1 \le x \le 6$. Also indicate the location (the value of $x$) of the global maximum and global minimum.
  
  Global maximum:
  (If rounding, keep at least four digits.)
  
  Location of global maximum: $x =$
  
  Global minimum:
  (If rounding, keep at least four digits.)
  
  Location of global minimum: $x= $

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Math 1241, Fall 2020