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<h1>🎯 Adventure 13 — Activity 1 (Solution)</h1>
<div class="subtitle">Completed derivatives and tables</div>
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<div class="left-col">
<h2>1) f(x) = x²</h2>
<p><b>f′(x)=</b> 2x <b>f″(x)=</b> 2</p>
<table class="gridtable">
<tr><th>x</th><th>f</th><th>f′</th><th>f″</th></tr>
<tr><td>-2</td><td>4</td><td>-4</td><td>2</td></tr>
<tr><td>-1</td><td>1</td><td>-2</td><td>2</td></tr>
<tr><td>0</td><td>0</td><td>0</td><td>2</td></tr>
<tr><td>1</td><td>1</td><td>2</td><td>2</td></tr>
<tr><td>2</td><td>4</td><td>4</td><td>2</td></tr>
</table>
<p><b>(0,0) is a minimum</b> because f′(0)=0 and f″(0)>0.</p>
<h2 style="margin-top:1.2rem;">2) f(x) = −x²</h2>
<p><b>f′(x)=</b> −2x <b>f″(x)=</b> −2</p>
<table class="gridtable">
<tr><th>x</th><th>f</th><th>f′</th><th>f″</th></tr>
<tr><td>-2</td><td>-4</td><td>4</td><td>-2</td></tr>
<tr><td>-1</td><td>-1</td><td>2</td><td>-2</td></tr>
<tr><td>0</td><td>0</td><td>0</td><td>-2</td></tr>
<tr><td>1</td><td>-1</td><td>-2</td><td>-2</td></tr>
<tr><td>2</td><td>-4</td><td>-4</td><td>-2</td></tr>
</table>
<p><b>(0,0) is a maximum</b> because f′(0)=0 and f″(0)<0.</p>
<h2 style="margin-top:1.2rem;">3) f(x) = x³</h2>
<p><b>f′(x)=</b> 3x² <b>f″(x)=</b> 6x</p>
<table class="gridtable">
<tr><th>x</th><th>f</th><th>f′</th><th>f″</th></tr>
<tr><td>-2</td><td>-8</td><td>12</td><td>-12</td></tr>
<tr><td>-1</td><td>-1</td><td>3</td><td>-6</td></tr>
<tr><td>0</td><td>0</td><td>0</td><td>0</td></tr>
<tr><td>1</td><td>1</td><td>3</td><td>6</td></tr>
<tr><td>2</td><td>8</td><td>12</td><td>12</td></tr>
</table>
<p><b>(0,0) is an inflection point</b> because the concavity changes (f″ changes sign).</p>
<h2 style="margin-top:1.2rem;">4) f(x) = −x³</h2>
<p><b>f′(x)=</b> −3x² <b>f″(x)=</b> −6x</p>
<table class="gridtable">
<tr><th>x</th><th>f</th><th>f′</th><th>f″</th></tr>
<tr><td>-2</td><td>8</td><td>-12</td><td>12</td></tr>
<tr><td>-1</td><td>1</td><td>-3</td><td>6</td></tr>
<tr><td>0</td><td>0</td><td>0</td><td>0</td></tr>
<tr><td>1</td><td>-1</td><td>-3</td><td>-6</td></tr>
<tr><td>2</td><td>-8</td><td>-12</td><td>-12</td></tr>
</table>
<p><b>(0,0) is an inflection point</b> because the concavity changes (f″ changes sign).</p>
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