Maximization and minimization

Math 201, Spring 19
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Total points: 3

We will explore the maxmima and minima of the function $f(x)=e^{- \frac{x^{4}}{24} - \frac{x^{3}}{18} + \frac{x^{2}}{2}}$. For this warm-up problem, we'll plot the function so you can see the answers even before doing the math.
1. Calculate the derivative of $f$.
  $f'(x)= $
  
  The first step in locating maxima and minima is finding critical points of $f$. Since the derivative exists everywhere, the only critical points are those where $f'(x)=0$.
  
  You should have calculated that $f'(x)$ is a polynomial times an exponential function (actually, a polynomial times $f$ itself). Can the exponential function ever be zero?
  . Therefore, the only critical points are where the polynomial is zero, i.e., where
  $= 0$.
  
  To find these critical points, factor out an $x$ from the polynomial (even better, factor out $-\frac{x}{6}$), so that the remaining factor is a quadratic polynomial. The factored equation becomes
  $= 0$.
  
  Now, factor the quadratic polynomial (or use the quadratic formula) to find all the critical points. Critical points: $x=$
  
  (Separate multiple answers by commas.)
2. Find the intervals where $f$ is increasing and decreasing. In this case, the critical points divide the number line into four intervals. In each interval, you can test $f'(x)$ for a point in the interval to determine if $f'(x)$ is positive or negative in the interval. From left to right, list the intervals and whether $f$ is increasing or decreasing in the interval.
  For the interval
  , $f$ is
  .
  For the interval
  , $f$ is
  .
  For the interval
  , $f$ is
  .
  For the interval
  , $f$ is
  .
  
  Need help?Hide help
  
  Hint
  Online, enter $\infty$ as oo or as the symbol ∞.
  Hide help
3. Local maxima can only occur where $f$ changes from increasing to decreasing (which can happen only at critical points). Find the local maxima of $f$.
  $f$ has local maxima at: $x=$
  and at $x=$
  
  (Enter in increasing order.)
  
  What values does $f(x)$ attain at the local maxima?
  $f\bigl($
  $\bigr)=$
  , $f\bigl($
  $\bigr)=$
  
  (Enter in same order as above. If rounding, keep at least 3 significant digits.)
4. Local minima can only occur where $f$ changes from decreasing to increasing (which can happen only at critical points). Find the local minima of $f$.
  $f$ has a local minimum at: $x=$
  .
  What value does $f(x)$ attain at the local minimum? $f\bigl($
  $\bigr)=$
5. Find the locations and values of the global maximum and minimum of $f$ in the interval $-1 \le x \le 3$. By global maximum and minimum, we mean the largest and smallest value of $f$ in that interval. The global maximum and minimum must occur at the critical points or the endpoints of the interval, so check the values of $f$ at those points to see which is the smallest and largest.
  Calculate the value of $f$ at the two critical points that occur in the interval. (Yes, you are repeating yourself.)
  $f\bigl($
  $\bigr)=$
  , $f\bigl($
  $\bigr)=$
  
  (Enter critical points in increasing order. If rounding, keep at least 3 significant digits.)
  
  Calculate the value of $f$ at the two endpoints.
  $f(-1)=$
  , $f(3)=$
  
  The global maximum of $f(x)$ in the interval $[-1,3]$ is the largest of those four values.
  The global maximum occurs at $x=$
  .
  The value of the global maximum is $f=$
  .
  
  The global minimum of $f(x)$ in the interval $[-1,3]$ is the smallest of those four values.
  The global minimum occurs at $x=$
  .
  The value of the global minimum is $f=$
  .
6. Find the locations and values of the global maximum and minimum of $f$ in the interval $-4 \le x \le 1$.
  Calculate the value of $f$ at the two critical points that occur in the interval.
  $f\bigl($
  $\bigr)=$
  , $f\bigl($
  $\bigr)=$
  
  (Enter critical points in increasing order. If rounding, keep at least 3 significant digits.)
  
  Calculate the value of $f$ at the two endpoints.
  $f(-4)=$
  , $f(1)=$
  
  The global maximum of $f(x)$ in the interval $[-4,1]$ is the largest of those four values.
  The global maximum occurs at $x=$
  .
  The value of the global maximum is $f=$
  .
  
  The global minimum of $f(x)$ in the interval $[-4,1]$ is the smallest of those four values.
  The global minimum occurs at $x=$
  .
  The value of the global minimum is $f=$
  .
Let $f(t)=t e^{- \frac{t}{2}}$.
1. Find the critical points of $f$.
  Critical points: $t=$
  . (Separate multiple answers by commas.)
2. Find the points $t$ where $f$ has local maxima and minima. (Local maxima and minima occur when $f$ switches between increasing and decreasing, which can only occur at critical points.) What are the values $f(t)$ of the local maxima and minima?
  $f(t)$ is increasing in the interval
  .
  $f(t)$ is decreasing in the interval
  
  Local maxima: $t=$
  . At local maxima, $f=$
  
  Local minima: $t=$
  . At local minima, $f=$
  
  Enter in increasing order, separating multiple answers by commas. If none, enter none. If rounding, include at least 3 significant digits.
  
  Need help?Hide help
  
  Hint
  Online, enter $\infty$ as oo or as the symbol ∞.
  Hide help
3. Find the locations and values of the global maximum and minimum of $f$ in the interval $1 \le t \le 4$.
  Checking the critical point: $f\bigl($
  $\bigr)=$
  
  Checking left endpoint: $f(1) = $
  
  Checking right endpoint $f(4) = $
  
  The global maximum of $f(t)$ in the interval $[1,4]$ occurs at $t=$
  .
  The value of the global maximum is $f=$
  .
  
  The global minimum of $f(t)$ in the interval $[1,4]$ occurs at $t=$
  .
  The value of the global minimum is $f=$
  .
Let $g$ be the function $g(y)=e^{- y^{3} + y}$.
1. Find the critical points of $g$.
  $y=$
  
  Separate multiple answers by commas.
  Need help?Hide help
  
  Hint
  Online, you can enter a square root such as $\sqrt(2)$ as sqrt(2) or you can round your answer to at least 3 significant digits.
  Hide help
2. Find the regions where $g$ is increasing and where $g$ is decreasing.
  $g(y)$ is increasing if
  $\lt y \lt $
  
  $g(y)$ is decreasing if $y \lt $
  or if $y \gt$
3. Find the locations and values of local maxima and minima of $g$.
  Local maxima occur at $y=$
  . Their values are $g=$
  
  Local minima occur at $y=$
  . Their values are $g=$
  
  Separate multiple answer by commas; list in order according to location.
4. Find the locations and values of the global maximum and minimum of $g$ on the interval $-1 \le y \le 2$.
  Global maximum occurs at $y=$
  . Its value is $g=$
  
  Global minimum occurs at $y=$
  . Its value is $g=$
Let $f$ be the function $f(x) =\left(2 x^{2} + 2 x + 1\right) e^{2 x}$.
1. Find the critical points of $f$: $x=$
2. Find the locations and values of local maxima and minima of $f$.
  Local maxima: $x=$
  .
  Local minima: $x=$
  .
  Enter in increasing order, separated by commas. If none, enter none.
3. Find the locations and values of the global maximum and minimum of $f$ on the interval $-2 \le x \le 0$.
  Global maximum occurs at $x=$
  . Its value is $f=$
  
  Global minimum occurs at $x=$
  . Its value is $f=$
  
  Need help?Hide help
  
  Hint
  If rounding, include at least 3 significant digits.
  Hide help
Let $m$ be the function $m(x)=x \left(- c + x\right)$, where $c$ is a positive parameter.
1. Find the locations and values of local maxima and minima of $m$.
  Local maxima: $x=$
  . Their values are $m=$
  
  Local minima: $x=$
  . Their values are $m=$
  .
  Enter in increasing order, separated by commas. If none, enter none.
  
  Need help?Hide help
  
  Hint
  Find the critical points. Remember to pretend $c$ is a number when taking the derivative. After setting the derivative to zero, solve for $x$. It's OK if the resulting critical point(s) depend on $c$.
  To determine whether any critical points found are local maxima or minima, you need to check the sign of $m(x)$ on the intervals divided by the critical points.
  
  If having $c$ as a parameter is too confusing, try replacing $c$ with some number, like 47. Once you get your answer for that value of $c$, try to figure out how to translate back to the generic parameter $c$.
  
  Hide help
2. Find the locations and values of the global maximum and minimum of $m$ on the interval $0 \le x \le 3c$.
  Global maximum occurs at $x=$
  . Its value is $m=$
  
  Global minimum occurs at $x=$
  . Its value is $m=$
  .
  
  Need help?Hide help
  
  Hint
  Check the value of $m$ at the endpoints and the critical points. Again, you could first try replacing $c$ with a specific value like 47 to get an idea how to do this.
  Hide help
Imagine that, if every year, a fraction $k$ of the fish are harvested from a lake, then the average number of fish in the lake will be $p(k)=1000 \left(1 - k\right)^{2}$. (We aren't going to model why this is true; we'll just imagine that, given how the fish reproduce, $p(k)$ is the equilibrium value of the fish population when a fraction $k$ are harvested each year.)
We want to determine the fraction of fish you should harvest to maximum the number of fish harvested each year. Let's call the actual harvest $h(k)$. If the steady state population size is $p(k)=1000 \left(1 - k\right)^{2}$ and you harvest a fraction $k$, how many fish do you harvest each year? $h(k)$ =

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

In this case, what will be the average number of fish in the lake?

How many fish will be harvested each year?

Need help?Hide help

Hint
This is clearly a maximization problem. What function should we maximize? $p(k)$? $h(k)$?
Do we want a local maximum or a global maximum?

What is the relevant range of fraction harvested $k$? (I.e., what numbers for $k$ make sense?)

Once you find the value of $k$ that maximizes the appropriate function, the values of both functions $p$ and $h$ at that optimal fraction are relevant.

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Math 201, Spring 19

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Math 201, Spring 19