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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.
   
   Solution
2. Below is the graph of some function, \(f\left( x \right)\). Identify all of the relative extrema and absolute extrema of the function.
   
   Solution
3. Sketch the graph of \(g\left( x \right) = {x^2} - 4x\) and identify all the relative extrema and absolute extrema of the function on each of the following intervals.
   1. \(\left( { - \infty ,\infty } \right)\)
   2. \(\left[ { - 1,4} \right]\)
   3. \(\left[ {1,3} \right]\)
   4. \(\left[ {3,5} \right]\)
   5. \(\left( { - 1,5} \right]\)
   Solution
4. Sketch the graph of \(h\left( x \right) = - {\left( {x + 4} \right)^3}\)and identify all the relative extrema and absolute extrema of the function on each of the following intervals.
   1. \(\left( { - \infty ,\infty } \right)\)
   2. \(\left[ { - 5.5, - 2} \right]\)
   3. \(\left[ { - 4, - 3} \right)\)
   4. \(\left[ { - 4, - 3} \right]\)
   Solution
5. Sketch the graph of some function on the interval \(\left[ {1,6} \right]\) that has an absolute maximum at \(x = 6\) and an absolute minimum at \(x = 3\). Solution
6. Sketch the graph of some function on the interval \(\left[ { - 4,3} \right]\) that has an absolute maximum at \(x = - 3\) and an absolute minimum at \(x = 2\). Solution
7. Sketch the graph of some function that meets the following conditions :
   1. The function is continuous.
   2. Has two relative minimums.
   3. One of relative minimums is also an absolute minimum and the other relative minimum is not an absolute minimum.
   4. Has one relative maximum.
   5. Has no absolute maximum.
   Solution