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<quiz>
  <question type="multichoice">
    <name>
      <text>10.2.20 (Stationary points and zeros)</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[Below are listed some possible behaviours of a differentiable function \(f\).&nbsp; Select all those behaviours that are indeed possible.<br>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>The function \(f(x) = e^x\) has no stationary points and no roots.</p><p></p><p>The function \(f(x) = x\) has no stationary points and exactly one root.</p><p></p><p>The function \(f(x) = x^2 + 1\) has exactly one stationary point and no roots.</p><p></p><p>The function \(f(x) = x^2\) has exactly one stationary point and exactly one root.</p><p></p><p>The function \(f(x) = x^2 -1\) has exactly one stationary point and two roots.</p><p>Between any two zeros of a differentiable function there must be a stationary point.&nbsp; We have not proved it but a sketch should make it clear.&nbsp; So two zeros but no stationary points, and three zeros and only one stationary point are both impossible behaviours.<br></p>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0.3333333</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <single>false</single>
    <shuffleanswers>false</shuffleanswers>
    <answernumbering>abc</answernumbering>
    <showstandardinstruction>1</showstandardinstruction>
    <correctfeedback format="html">
      <text>Your answer is correct.</text>
    </correctfeedback>
    <partiallycorrectfeedback format="html">
      <text>Your answer is partially correct.</text>
    </partiallycorrectfeedback>
    <incorrectfeedback format="html">
      <text>Your answer is incorrect.</text>
    </incorrectfeedback>
    <shownumcorrect/>
    <answer fraction="20" format="html">
      <text><![CDATA[<p>\(f\) has no stationary points and no roots.<br></p>]]></text>
      <feedback format="html">
        <text/>
      </feedback>
    </answer>
    <answer fraction="20" format="html">
      <text><![CDATA[<p>\(f\) has no stationary points and exactly one root.<br></p>]]></text>
      <feedback format="html">
        <text/>
      </feedback>
    </answer>
    <answer fraction="0" format="html">
      <text><![CDATA[<p>\(f\) has no stationary points and exactly two roots.<br></p>]]></text>
      <feedback format="html">
        <text/>
      </feedback>
    </answer>
    <answer fraction="20" format="html">
      <text><![CDATA[<p>\(f\) has exactly one stationary point and no roots.<br></p>]]></text>
      <feedback format="html">
        <text/>
      </feedback>
    </answer>
    <answer fraction="20" format="html">
      <text><![CDATA[<p>\(f\) has exactly one stationary point and exactly one root.<br></p>]]></text>
      <feedback format="html">
        <text/>
      </feedback>
    </answer>
    <answer fraction="20" format="html">
      <text><![CDATA[<p>\(f\) has exactly one stationary point and exactly two roots.<br></p>]]></text>
      <feedback format="html">
        <text/>
      </feedback>
    </answer>
    <answer fraction="0" format="html">
      <text><![CDATA[<p>\(f\) has exactly one stationary point and exactly three roots.<br></p>]]></text>
      <feedback format="html">
        <text/>
      </feedback>
    </answer>
  </question>
</quiz>
