Chapter 6.4

Notes

Linguistic Mapping Exercises

Knowledge Checks

  1. For each function \(f\) given below, compute the derivative of \(f\) to determine where \(f\) is increasing and decreasing:
    1. \(f(x) = \ln(x-1)\) and \(I = (1, \infty)\)
    2. \(f(x)= x^2+3x\)
    3. \(f(x) = -x(x+5)(x-2)\)
  2. Take \(f\) to be a differentiable function on \([-6, 3.5]\). Given this sketch of \(f^\prime\) below, find and classify all extremal points of \(f\) in \((-6, 3.5)\). And find where \(f\) is increasing and decreasing in \((-6,3.5).\)

  1. Take \(f\) to be a function \(f\) with \(f^\prime(x) = -(x+3)^2(x+2)(x-3)^4(x-6)^3\). Determine all points in \(\mathbb R\) where \(f\) has a local maximum or minimum.
  2. Find and classify all extremal points for \(f(x)= \tfrac{1}{3}x^3-2x^2-5x\) on \([-2,6].\)
  3. For each function \(f\) that is given below, determine \(f^{\prime\prime}(x):\)
    1. \(f(x) = 4^x+x^5+\mathrm{e}^2\)
    2. \(f(x) = \frac{1}{x-2}\ln(x)\)
  4. Take \(f(x) = -\tfrac{1}{3} x^3-x^2 +3x+4\). Find all inflection points of \(f\) and give all maximal intervals on which \(f\) is convex or concave.
  5. Sketch an example of a continuous function \(f\) that has the property that:
    • \(f^{\prime\prime}\) is positive on \((-\infty, -2)\cup(0, \infty),\)
    • \(f^{\prime\prime}\) is negative on \((-2, 0),\)
    • the zero set of \(f^{\prime\prime}\) is \(\{-2,0\},\)
    • \(f^\prime\) is negative on \((-\infty, -3)\cup(-1.5,1.25),\)
    • \(f^\prime\) is positive on \((-3, -1.5)\cup(1.25,\infty),\)
    • the zero set of \(f^{\prime}\) is \(\{-3,-1.5,1.25\}.\)
  6. Sketch the function \(f(x) = -\tfrac{1}{3} x^3-x^2 +3x+4\) by determining where \(f\) is increasing, decreasing, convex, concave, and where \(f\) has a local max, local min, and inflection points.

Remember, do the knowledge checks before checking the solutions.

Practice Problems

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