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    <name>
      <text>18.3.2.4 T Locate stationary points (copy)</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Locate the stationary points of</p>
\[f(x,y) = {@f@}.\]
<p>First find the partial derivatives of \(f\):</p>
<p>\(\displaystyle \frac{\partial f}{\partial x} = \) [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</p>
<p>\(\displaystyle \frac{\partial f}{\partial y} = \) [[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</p>
<p>Now solve simultaenously the equations \(\displaystyle \frac{\partial f}{\partial x} = 0\) and \(\displaystyle \frac{\partial f}{\partial y} = 0\).
</p>
<p><b>Note:</b> write your answer as a set of points. For example, if you found the stationary points \((0,2)\) and \((1,3)\), your input should be <code>{[0,2], [1,3]}</code>.</p>
<p>Stationary points: [[input:ans3]] [[validation:ans3]] [[feedback:prt3]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>We are asked to find the stationary points of \(f(x,y) = {@f@}\).</p>
<p>We begin by finding the partial derivatives of \(f\):</p>
\[\begin{aligned}
    \frac{\partial f}{\partial x} &amp;= {@ta1@} \\
    \frac{\partial f}{\partial y} &amp;= {@ta2@}
\end{aligned}\]
<p>Next, we find the points where both partial derivatives are zero. Note that each of them
    only depends on one variable, so that we can solve each equation separately.
</p>
\[\begin{aligned}
    \frac{\partial f}{\partial x} = {@ta1@} = 0 &amp;\implies x = \pm {@x1@} \\
    \frac{\partial f}{\partial y} = {@ta2@} = 0 &amp;\implies y = {@y1@}
\end{aligned}\]
<p>Therefore, there are two points where both partial derivatives are zero: \(({@x1@},{@y1@})\)
    and \(({@x2@},{@y1@})\). These are the <strong>stationary points</strong>.
</p>]]></text>
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    <idnumber/>
    <stackversion>
      <text>2021120900</text>
    </stackversion>
    <questionvariables>
      <text>ordergreat(x, y);
/* Randomisation */
/* First find zeroes of partial derivatives */
x1: rand_with_step(1, 4, 1);
x2: -x1;
y1: rand_with_prohib(-4, 4, [0]);
/* Then construct partial derivatives */
f_x: expand(3 * (x - x1) * (x - x2));
f_y: expand (2 * (y - y1));
/* Lastly, construct f */
f: int(f_x, x) + int(f_y, y) + rand_with_prohib(-5, 5, [0]);

ta1: f_x;
ta2: f_y;
ta3: {[x1, y1], [x2, y1]};</text>
    </questionvariables>
    <specificfeedback format="html">
      <text/>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\(f(x,y) = {@f@}, ({@x1@},{@y1@}), ({@x2@},{@y1@})\)</text>
    </questionnote>
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      <text/>
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    <prtcorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
    </prtcorrect>
    <prtpartiallycorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
    </prtpartiallycorrect>
    <prtincorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
    </prtincorrect>
    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
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