Math Insight

Critical points, maximization and minimization problems

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  1. Let $f(x) =e^{- \frac{x^{3}}{6} + \frac{3 x^{2}}{4} + 2} $.
    1. Calculate $f'(x)$.
    2. Find the critical points of $f$.
    3. Determine on which intervals $f$ is increasing and $f$ is decreasing.
    4. Sketch the graph of $f(x)$. Your sketch should be consistent with the information you determined.

  2. Let $f(x) =e^{\frac{2 x^{3}}{3} + 6 x^{2} + 18 x + 3} $.
    1. Calculate $f'(x)$.
    2. Find the critical points of $f$.

    3. Find the local minima and maxima (or local extrema) of $f$. For each extremum, determine three things: its location (i.e., value of $x$), its value (i.e., value of $f(x)$), and whether it is a local maximum or a local minimum.

    4. Find the global maximum and global minimum of the function $f(x)$ on the interval $-4 \le x \le -2$. Also indicate the location (the value of $x$) of the global maximum and global minimum.

    5. Sketch the graph of $f(x)$ on the interval $-4 \le x \le -2$. Your sketch should be consistent with the information you determined.

  3. Let $f(x) =10 \left(x^{2} + 5 x - 6\right)^{4} - 3 $.
    1. Calculate $f'(x)$
    2. Find the critical points of $f$.

    3. Find the local minima and maxima (or local extrema) of $f$. For each extremum, determine three things: its location (i.e., value of $x$), its value (i.e., value of $f(x)$), and whether it is a local maximum or a local minimum.

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

    5. Sketch the graph of $f(x)$ on the interval $-5 \le x \le -1$. Your sketch should be consistent with the information you determined.

  4. Let $f(x) =\left(3 x^{2} - 9 x\right) e^{\frac{x}{2}} $.
    1. Calculate $f'(x)$.
    2. Find the critical points of $f$.

    3. Find the local minima and maxima (or local extrema) of $f$. For each extremum, determine three things: its location (i.e., value of $x$), its value (i.e., value of $f(x)$), and whether it is a local maximum or a local minimum.

    4. Find the global maximum and global minimum of the function $f(x)$ on the interval $-6 \le x \le 3$. Also indicate the location (the value of $x$) of the global maximum and global minimum.

    5. Sketch the graph of $f(x)$ on the interval $-6 \le x \le 3$. Your sketch should be consistent with the information you determined.

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Math 201, Spring 2016