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Section 4-3 : Minimum and Maximum Values

  1. Below is the graph of some function, \(f\left( x \right)\). Identify all of the relative extrema and absolute extrema of the function.
    This graph has five line segments to it.  The first segment starts at (-10,6) and ends at (-6,-5).  The second segment starts at (-6,-5) and ends at (-2,4).  The third segment starts at (-2,4) and ends at (2,-1).  The fourth segment starts at (2,-1) and ends at (4,8).  The final segment starts at (4,8) and ends at (7,-6).
  2. Below is the graph of some function, \(f\left( x \right)\). Identify all of the relative extrema and absolute extrema of the function.
    This graph starts at (-4, -8) and increases until it reaches a peak at (-3,0).  It then decreases until it reaches a valley at (-2,-3).  The curve then increases to a peak at (1,9) and then decreases to a valley at (5.5, -5).  Finally, it increases until it reaches (7,3).
  3. Below is the graph of some function, \(f\left( x \right)\). Identify all of the relative extrema and absolute extrema of the function.
    This graph starts at (-3, -3) and increases until it reaches a peak at (-1,1.5).  It then decreases until it reaches a valley at (0,1).  The curve then increases to a peak at (1,1.5) and then decreases until it reaches (3,-3).
  4. Below is the graph of some function, \(f\left( x \right)\). Identify all of the relative extrema and absolute extrema of the function.
    This graph starts at (-10, -8) and increases until it reaches a peak at (-6,8) and then decreases until it hits (-4,4).  The graph then jumps down to the point (-4,-2) and increases until it reaches the point (2,10).  The graph then jumps down to the point (2,6) and decreases until it reaches a valley at (4,-10) and then increases until it reaches (5,-6).
  5. Sketch the graph of \(f\left( x \right) = 3 - {\frac{1}{2}}x\) and identify all the relative extrema and absolute extrema of the function on each of the following intervals.
    1. \(\left( { - \infty ,\infty } \right)\)
    2. \(\left[ { - 3,2} \right]\)
    3. \(\left[ { - 4,1} \right)\)
    4. \(\left( {0,5} \right)\)
  6. Sketch the graph of \(g\left( x \right) = {\left( {x - 2} \right)^2} + 1\) and identify all the relative extrema and absolute extrema of the function on each of the following intervals.
    1. \(\left( { - \infty ,\infty } \right)\)
    2. \(\left[ {0,3} \right]\)
    3. \(\left[ { - 1,5} \right]\)
    4. \(\left[ { - 1,1} \right]\)
    5. \(\left[ {1,3} \right)\)
    6. \(\left( {2,4} \right)\)
  7. Sketch the graph of \(h\left( x \right) = {{\bf{e}}^{3 - x}}\) and identify all the relative extrema and absolute extrema of the function on each of the following intervals.
    1. \(\left( { - \infty ,\infty } \right)\)
    2. \(\left[ { - 1,3} \right]\)
    3. \(\left[ { - 6, - 1} \right]\)
    4. \(\left( {1,4} \right]\)
  8. Sketch the graph of \(h\left( x \right) = \cos \left( x \right) + 2\) and identify all the relative extrema and absolute extrema of the function on each of the following intervals. Do, all work for this problem in radians.
    1. \(\left( { - \infty ,\infty } \right)\)
    2. \(\left[ {\displaystyle - \frac{\pi }{3},\frac{\pi }{4}} \right]\)
    3. \(\left[ {\displaystyle - \frac{\pi }{2},2\pi } \right]\)
    4. \(\left[ {\displaystyle \frac{1}{2},1} \right]\)
  9. Sketch the graph of a function on the interval \(\left[ {3,9} \right]\) that has an absolute maximum at \(x = 5\) and an absolute minimum at \(x = 4\).
  10. Sketch the graph of a function on the interval \(\left[ {0,10} \right]\) that has an absolute minimum at \(x = 5\) and an absolute maximums at \(x = 0\) and \(x = 10\).
  11. Sketch the graph of a function on the interval \(\left( { - \infty ,\infty } \right)\) that has a relative minimum at \(x = - 7\), a relative maximum at \(x = 2\) and no absolute extrema.
  12. Sketch the graph of a function that meets the following conditions :
    1. Has at least one absolute maximum.
    2. Has one relative minimum.
    3. Has no absolute minimum.
  13. Sketch the graph of a function that meets the following conditions :
    1. Graphed on the interval \(\left[ {2,9} \right]\).
    2. Has a discontinuity at some point interior to the interval.
    3. Has an absolute maximum at the discontinuity in part (b).
    4. Has an absolute minimum at the discontinuity in part (b).
  14. Sketch the graph of a function that meets the following conditions :
    1. Graphed on the interval \(\left[ { - 4,10} \right]\).
    2. Has no relative extrema.
    3. Has an absolute maximum at one end point.
    4. Has an absolute minimum at the other end point.
  15. Sketch the graph of a function that meets the following conditions :
    1. Has a discontinuity at some point.
    2. Has an absolute maximum and an absolute minimum.
    3. Neither absolute extrema occurs at the discontinuity.