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Application of Derivatives
NCERT Exemplar Class 12
CBSE
Grade 12
Question:
Show that height of cylinder of maximum volume that can be inscribed in a sphere of radius $R$ is $\dfrac{2R}{\sqrt{3}}$.
Step-by-Step Solution
Given: Problem statement: Show that height of cylinder of maximum volume that can be inscribed in a sphere of radius $R$ is $\dfrac{2R}{\sqrt{3}}$. Step 1: Formulate single-variable objective function: Express objective quantity (volume, area, distance, profit) in terms of single variable. [1.0 Mark] Step 2: Differentiate and find critical points: Compute first derivative $f'(x) = 0$ and solve for critical points. [1.0 Mark] Step 3: Verify maxima/minima using Second Derivative Test: Check $f''(x) < 0$ for maxima or $f''(x) > 0$ for minima. [1.0 Mark] Conclusion: Maximum/minimum dimensions proved.
--- 🎯 Official CBSE Marking Scheme: Formulating objective function: 1.0 Mark Evaluating derivative and critical points: 1.0 Mark Second derivative test and final dimensions: 1.0 Mark
Correct Answer:
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