Let

$\displaystyle f(x) = {{2\,x^3}\over{3}}-3\,x^2+4\,x-2 $

Then answer the following questions.

  1. Find the critical numbers of $f(x)$.

    $x = 1 $, $2$ $x = 0 $, $3$ $x = -3 $, $0$ $x = -2 $, $-1$

  2. Choose the correct table for I/D test.

    Interval $x < 1$ $1 < x < 2$ $2 < x $
    $f'(x)$ pos neg pos

    Interval $x < 1$ $1 < x < 2$ $2 < x $
    $f'(x)$ neg pos neg

    Interval $x < 0$ $0 < x < 3$ $3 < x $
    $f'(x)$ neg pos neg

    Interval $x < 0$ $0 < x < 3$ $3 < x $
    $f'(x)$ pos neg pos

  3. Choose the correct statement.

    $y = -{{1}\over{3}}$ is local maximum at $x = 1 $ . $y = -2$ is local maximum at $x = 0 $ . $y = 1$ is local minimum at $x = 3$ . $y = -{{2}\over{3}}$ is local maximum at $x = 2$ .

  4. Find $\displaystyle\lim_{x\to -\infty}f(x)$ and $\displaystyle\lim_{x\to \infty}f(x)$ for horizontal asymptotes.

    $\displaystyle\lim_{x\to -\infty}f(x) =$ $\displaystyle -\infty $ and $\displaystyle\lim_{x\to \infty}f(x) =$ $\displaystyle \infty $

    $\displaystyle\lim_{x\to -\infty}f(x) =$ $\displaystyle \infty $ and $\displaystyle\lim_{x\to \infty}f(x) =$ −2

    $\displaystyle\lim_{x\to -\infty}f(x) =$ $\displaystyle -\infty $ and $\displaystyle\lim_{x\to \infty}f(x) =$ −2

    $\displaystyle\lim_{x\to -\infty}f(x) =$ $\displaystyle \infty $ and $\displaystyle\lim_{x\to \infty}f(x) =$ $\displaystyle -\infty $

  5. Sketch the graph of $y = f(x)$.

    Image s20quiz16_pick_plot0

    Image s20quiz16_pick_plot1

    Image s20quiz16_pick_plot2

    Image s20quiz16_pick_plot3


Department of Mathematics
Last modified: 2025-01-31