Math Insight

Critical points, maximization, and minimization practice

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  1. Let $f(x) =- 5 \left(x^{2} + 12 x + 20\right)^{2} + 3 $.

    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?

    4. Find the global maximum and global minimum of the function $f(x)$ on the interval $-11 \le x \le -1$.

      Global maximum:
      (If rounding, keep at least four digits.)

      Global minimum:
      (If rounding, keep at least four digits.)

  2. Let $f(x) =3\ln(6x) - \sqrt{ 3 x} $.

    1. Calculate $f'(x)$.

      $f'(x) = $

    2. The function $f(x)$ and its derivative $f'(x)$ are defined only for positive $x$, so we will consider only $x > 0$. 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=

      (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.)

    4. Find the global maximum and global minimum of the function $f(x)$ on the interval $11 \le x \le 17$. 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= $

  3. Let $f(x) =e^{x^{3} + 6 x^{2} + 12 x - 4} $.

    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?

    4. Find the global maximum and global minimum of the function $f(x)$ on the interval $-5 \le x \le 1$. 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= $

  4. After a nearby chemical spill, the pollution level of a bay is monitored. The pollution level $t$ weeks after the spill is given $p(t)=9 t^{4} e^{- t}$.

    For how long after the spill does the pollution level in the bay continue to rise? What is the maximum pollution level?

    The pollution level continues to rise for
    weeks.

    The maximum pollution level is
    . (If rounding, keep at least four digits.)

  5. Let $f(x) =- \left(x - 3\right) \left(x - 2\right) \left(3 x - 1\right) = - 3 x^{3} + 16 x^{2} - 23 x + 6$.

    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 three intervals. What are these three intervals? Enter them from left to right. Enter oo for $\infty$.

      Interval 1 =

      Interval 2 =

      Interval 3 =

    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

      On interval 3, $f$ is

    5. What are the roots of $f$ itself, i.e., at what points is $f(x)=0$?

      Roots =
      (Separate answers by commas, do not round.)

    6. Using this information, sketch the graph of $f(x)$.

      We can't grade this automatically, so you'll just have to check it yourself, either by checking the solution or by graphing the function using a computer or calculator.)

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Math 201, Spring 2016