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<h1>🎯 Adventure 13 — Activity 1 (Solution)</h1>
<div class="subtitle">Completed student page with the answers filled in</div>
<div class="two-col">
<div class="left-col">
<div class="callout">
<b>Goal:</b> For each function, compute <code>f</code>, <code>f′</code>, and <code>f″</code> at <code>x = −2, −1, 0, 1, 2</code>.
Then decide: does the point (0,0) give a <b>maximum</b>, <b>minimum</b>, or an <b>inflection point</b>?
</div>
<h2>1) f(x) = x²</h2>
<p>Compute derivatives:</p>
<p><b>f′(x) =</b> <input readonly id="d1" placeholder="e.g., 2x" style="max-width:220px;" type="text" value="2x"/></p>
<p><b>f″(x) =</b> <input readonly id="d2" placeholder="e.g., 2" style="max-width:220px;" type="text" value="2"/></p>
<table class="gridtable">
<tr><th>x</th><th>f</th><th>f′</th><th>f″</th></tr>
<tr><td>-2</td><td><input readonly id="x2_f_-2" step="any" type="number" value="4"/></td><td><input readonly id="x2_fp_-2" step="any" type="number" value="-4"/></td><td><input readonly id="x2_fpp_-2" step="any" type="number" value="2"/></td></tr>
<tr><td>-1</td><td><input readonly id="x2_f_-1" step="any" type="number" value="1"/></td><td><input readonly id="x2_fp_-1" step="any" type="number" value="-2"/></td><td><input readonly id="x2_fpp_-1" step="any" type="number" value="2"/></td></tr>
<tr><td>0</td><td><input readonly id="x2_f_0" step="any" type="number" value="0"/></td><td><input readonly id="x2_fp_0" step="any" type="number" value="0"/></td><td><input readonly id="x2_fpp_0" step="any" type="number" value="2"/></td></tr>
<tr><td>1</td><td><input readonly id="x2_f_1" step="any" type="number" value="1"/></td><td><input readonly id="x2_fp_1" step="any" type="number" value="2"/></td><td><input readonly id="x2_fpp_1" step="any" type="number" value="2"/></td></tr>
<tr><td>2</td><td><input readonly id="x2_f_2" step="any" type="number" value="4"/></td><td><input readonly id="x2_fp_2" step="any" type="number" value="4"/></td><td><input readonly id="x2_fpp_2" step="any" type="number" value="2"/></td></tr>
</table>
<h2 style="margin-top:1.2rem;">2) f(x) = −x²</h2>
<p><b>f′(x) =</b> <input readonly id="d3" placeholder="e.g., -2x" style="max-width:220px;" type="text" value="−2x"/></p>
<p><b>f″(x) =</b> <input readonly id="d4" placeholder="e.g., -2" style="max-width:220px;" type="text" value="−2"/></p>
<table class="gridtable">
<tr><th>x</th><th>f</th><th>f′</th><th>f″</th></tr>
<tr><td>-2</td><td><input readonly id="nx2_f_-2" step="any" type="number" value="-4"/></td><td><input readonly id="nx2_fp_-2" step="any" type="number" value="4"/></td><td><input readonly id="nx2_fpp_-2" step="any" type="number" value="-2"/></td></tr>
<tr><td>-1</td><td><input readonly id="nx2_f_-1" step="any" type="number" value="-1"/></td><td><input readonly id="nx2_fp_-1" step="any" type="number" value="2"/></td><td><input readonly id="nx2_fpp_-1" step="any" type="number" value="-2"/></td></tr>
<tr><td>0</td><td><input readonly id="nx2_f_0" step="any" type="number" value="0"/></td><td><input readonly id="nx2_fp_0" step="any" type="number" value="0"/></td><td><input readonly id="nx2_fpp_0" step="any" type="number" value="-2"/></td></tr>
<tr><td>1</td><td><input readonly id="nx2_f_1" step="any" type="number" value="-1"/></td><td><input readonly id="nx2_fp_1" step="any" type="number" value="-2"/></td><td><input readonly id="nx2_fpp_1" step="any" type="number" value="-2"/></td></tr>
<tr><td>2</td><td><input readonly id="nx2_f_2" step="any" type="number" value="-4"/></td><td><input readonly id="nx2_fp_2" step="any" type="number" value="-4"/></td><td><input readonly id="nx2_fpp_2" step="any" type="number" value="-2"/></td></tr>
</table>
<h2 style="margin-top:1.2rem;">3) f(x) = x³</h2>
<p><b>f′(x) =</b> <input readonly id="d5" placeholder="e.g., 3x²" style="max-width:220px;" type="text" value="3x²"/></p>
<p><b>f″(x) =</b> <input readonly id="d6" placeholder="e.g., 6x" style="max-width:220px;" type="text" value="6x"/></p>
<table class="gridtable">
<tr><th>x</th><th>f</th><th>f′</th><th>f″</th></tr>
<tr><td>-2</td><td><input readonly id="x3_f_-2" step="any" type="number" value="-8"/></td><td><input readonly id="x3_fp_-2" step="any" type="number" value="12"/></td><td><input readonly id="x3_fpp_-2" step="any" type="number" value="-12"/></td></tr>
<tr><td>-1</td><td><input readonly id="x3_f_-1" step="any" type="number" value="-1"/></td><td><input readonly id="x3_fp_-1" step="any" type="number" value="3"/></td><td><input readonly id="x3_fpp_-1" step="any" type="number" value="-6"/></td></tr>
<tr><td>0</td><td><input readonly id="x3_f_0" step="any" type="number" value="0"/></td><td><input readonly id="x3_fp_0" step="any" type="number" value="0"/></td><td><input readonly id="x3_fpp_0" step="any" type="number" value="0"/></td></tr>
<tr><td>1</td><td><input readonly id="x3_f_1" step="any" type="number" value="1"/></td><td><input readonly id="x3_fp_1" step="any" type="number" value="3"/></td><td><input readonly id="x3_fpp_1" step="any" type="number" value="6"/></td></tr>
<tr><td>2</td><td><input readonly id="x3_f_2" step="any" type="number" value="8"/></td><td><input readonly id="x3_fp_2" step="any" type="number" value="12"/></td><td><input readonly id="x3_fpp_2" step="any" type="number" value="12"/></td></tr>
</table>
<h2 style="margin-top:1.2rem;">4) f(x) = −x³</h2>
<p><b>f′(x) =</b> <input readonly id="d7" placeholder="e.g., -3x²" style="max-width:220px;" type="text" value="−3x²"/></p>
<p><b>f″(x) =</b> <input readonly id="d8" placeholder="e.g., -6x" style="max-width:220px;" type="text" value="−6x"/></p>
<table class="gridtable">
<tr><th>x</th><th>f</th><th>f′</th><th>f″</th></tr>
<tr><td>-2</td><td><input readonly id="nx3_f_-2" step="any" type="number" value="8"/></td><td><input readonly id="nx3_fp_-2" step="any" type="number" value="-12"/></td><td><input readonly id="nx3_fpp_-2" step="any" type="number" value="12"/></td></tr>
<tr><td>-1</td><td><input readonly id="nx3_f_-1" step="any" type="number" value="1"/></td><td><input readonly id="nx3_fp_-1" step="any" type="number" value="-3"/></td><td><input readonly id="nx3_fpp_-1" step="any" type="number" value="6"/></td></tr>
<tr><td>0</td><td><input readonly id="nx3_f_0" step="any" type="number" value="0"/></td><td><input readonly id="nx3_fp_0" step="any" type="number" value="0"/></td><td><input readonly id="nx3_fpp_0" step="any" type="number" value="0"/></td></tr>
<tr><td>1</td><td><input readonly id="nx3_f_1" step="any" type="number" value="-1"/></td><td><input readonly id="nx3_fp_1" step="any" type="number" value="-3"/></td><td><input readonly id="nx3_fpp_1" step="any" type="number" value="-6"/></td></tr>
<tr><td>2</td><td><input readonly id="nx3_f_2" step="any" type="number" value="-8"/></td><td><input readonly id="nx3_fp_2" step="any" type="number" value="-12"/></td><td><input readonly id="nx3_fpp_2" step="any" type="number" value="-12"/></td></tr>
</table>
<div class="callout" style="margin-top:1rem;">
<b>Finish:</b> Use your tables to complete these statements:
<ul>
<li>For <code>x²</code>, we have a <b>minimum</b> at (0,0) because <code>f′(0)=</code> <b>0</b> and <code>f″(0)</code> is <b>positive</b>.</li>
<li>For <code>−x²</code>, we have a <b>maximum</b> at (0,0) because <code>f′(0)=</code> <b>0</b> and <code>f″(0)</code> is <b>negative</b>.</li>
<li>For <code>x³</code>, we have an <b>inflection point</b> at (0,0) because <code>f″(0)=</code> <b>0</b> and the concavity changes.</li>
<li>For <code>−x³</code>, we have an <b>inflection point</b> at (0,0) because <code>f″(0)=</code> <b>0</b> and the concavity changes.</li>
</ul>
</div>
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