Applications of Derivatives
Derivatives and Graphs
Grade 12
Question:
<p>Which of the following could be the sketch of graph of \(y = \frac{d}{dx}(x \log x)\)?</p>
<p>(a) Graph with minimum at \(x = 1/e\)</p>
<p>(b) Graph crossing x-axis</p>
<p>(c) Graph with specific shape</p>
<p>(d) Graph with specific shape</p>
Step-by-Step Solution
Key Concept: Differentiate $x\log x$ using product rule to get $\log x + 1$, then sketch this logarithmic curve.
<p><strong>Step 1:</strong> Find $\frac{d}{dx}(x \log x) = \log x + x \cdot \frac{1}{x} = \log x + 1$</p><p><strong>Step 2:</strong> This is a logarithmic function shifted up by 1 unit, which has a minimum value at $x = 1/e$ where $y = 0$.</p><p>∴ Answer is (a).</p>
Correct Answer: A