Applications of Derivatives
Lagrange's Mean Value Theorem
Grade 12
Question:
<p>A value of <i>C</i> for which the conclusion of the mean value theorem holds for the function <i>f</i>(<i>x</i>) = log<sub>e</sub><i>x</i> on the interval [1, 3] is</p>
<p>(a) \(2 \log_3 e\)</p>
<p>(b) \(\frac{1}{2}\log_e 3\)</p>
<p>(c) \(\log_3 e\)</p>
<p>(d) \(\log_e 3\)</p>
Step-by-Step Solution
Key Concept: Apply the Lagrange's mean value theorem formula to find the value of c where the derivative equals the average rate of change.
<p><strong>Using mean value theorem:</strong></p><p>$$f'(c) = \frac{f(3) - f(1)}{3 - 1}$$</p><p>$$\frac{1}{c} = \frac{\log_e 3 - \log_e 1}{2} = \frac{\log_e 3}{2}$$</p><p>$$c = \frac{2}{\log_e 3} = 2 \log_3 e$$</p><p>∴ Answer is (a).</p>
Correct Answer: A