Math Insight

Critical points, maximization and minimization problems

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  1. Let $f(x) =- x^{5} e^{- 3 x} $.
    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 $\frac{5}{12} \le x \le 3$. 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 $\frac{5}{12} \le x \le 3$. Your sketch should be consistent with the information you determined.

  2. Let $f(x) =e^{\frac{x^{3}}{3} - x^{2} + x - 3} $.
    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.

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

  4. Let $f(x) =-2\ln(2x) + \sqrt{ 4 x} $.
    1. Calculate $f'(x)$.
    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 $f$.
    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, and should only include positive values of $x$.