Math Insight

Minimization and maximization problems

 

Problem 1

Let $f$ be the function $f(x)=x^2e^x$.

  1. Find the critical points.
  2. Find the regions where $f$ is increasing and where $f$ is decreasing.
  3. Find the local maxima and minima of $f$.
  4. Find the global maximum and minimum of $f$ on the interval $-3 \le x \le 1$.

Problem 2

Let $g$ be the function $g(y)=e^{y-y^3}$.

  1. Find the critical points of $g$.
  2. Find the regions where $g$ is increasing and where $g$ is decreasing.
  3. Find the local maxima and minima of $g$.
  4. Find the global maximum and minimum of $g$ on the interval $-1 \le y \le 2$.

Problem 3

Let $h$ be the function $h(z)=(z-a)^2$ where $a$ is a parameter.

  1. Find the critical points of $h$.
  2. Find the regions where $h$ is increasing and where $h$ is decreasing.
  3. Find the local maxima and minima of $h$.
  4. Assume $a>0$. Find the global maximum and minimum of $h$ on the interval $0 \le z \le 3a$.

Problem 4

Let $k$ be the function $k(q)=\frac{q^2}{b^2}e^{-q}$, where $b$ is a parameter.

  1. Find the critical points of $k$.
  2. Find the regions where $k$ is increasing and where $k$ is decreasing.
  3. Find the local maxima and minima of $k$.
  4. Find the global maximum and minimum of $k$ on the interval $0 \le q \le 4$.

Problem 5

Let $m$ be the function $m(x)=x(x-c)$, where $c$ is a positive parameter.

  1. Find the critical points of $m$.
  2. Find the regions where $m$ is increasing and where $m$ is decreasing.
  3. Find the local maxima and minima of $m$.
  4. Find the global maximum and minimum of $m$ on the interval $0 \le x \le 3c$.

Problem 6

Let $n$ be the function $n(x)=x(x+c)$, where $c$ is a positive parameter.

  1. Find the critical points of $n$.
  2. Find the regions where $n$ is increasing and where $n$ is decreasing.
  3. Find the local maxima and minima of $n$.
  4. Find the global maximum and minimum of $n$ on the interval $-3c \le x \le 0$.

Problem 7

After a nearby chemical spill, the pollution level of a bay is monitored. The pollution level $t$ weeks after the spill is given $p(t)=8t^3e^{-2t}$.

  1. For how long after the spill does the pollution level in the bay continue to rise?
  2. When is the maximum pollution level reached? What is the maximum pollution level?

Problem 8

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 $p(k)=1000(1-k)^2$. This means that the annual fish harvest is $k \cdot p(k) = 1000 k (1-k)^2$. 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?

Problem 9

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 $p(k)=c(1-k)^2$, where $c$ is a positive number. This means that the annual fish harvest is $k \cdot p(k) = c k (1-k)^2$. 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?

One you have worked on a few problems, you can compare your solutions to the ones we came up with.

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