Math Insight

Critical points, maximization and minimization problems

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  1. Let $f(x) =e^{- \frac{x^{3}}{2} + \frac{9 x^{2}}{2} - \frac{27 x}{2} + 1} $.
    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) =-2\ln(4x) + \sqrt{ 6 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. 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 $\frac{5}{3} \le x \le \frac{20}{3}$. 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 $\frac{5}{3} \le x \le \frac{20}{3}$. Your sketch should be consistent with the information you determined.

  3. Let $f(x) =\ln{\left (x^{2} + 3 \right )}$. (Note that $f$ is defined for all $x$ since $x^{2} + 3$ is always positive.)
    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 $-3 \le x \le 2$. 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 $-3 \le x \le 2$. Your sketch should be consistent with the information you determined.

  4. Let $f(x) =- \left(x - 3\right) \left(x + 2\right) \left(3 x - 1\right) = - 3 x^{3} + 4 x^{2} + 17 x - 6$.
    1. Calculate $f'(x)$.
    2. Find the critical points of $f$.
    3. Determine on which intervals $f$ is increasing and $f$ is decreasing.
    4. What are the roots of $f$ itself, i.e., at what points is $f(x)=0$?
    5. Sketch the graph of $f(x)$. Your sketch should be consistent with the information you determined.

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Math 1241, Fall 2025