Question:

Given: Problem statement: An open tank with a square base and vertical sides is to be constructed from a metal sheet of given area $C^2$. Show that cost of construction will be minimum when depth of tank is half of its width.
Step 1: Set up geometric relations and constraint equation:
Relate variables using given perimeter/surface area constraint. [1.0 Mark]
Step 2: Express objective function in single variable:
Substitute constraint to get objective function $f(x)$. [1.0 Mark]
Step 3: Differentiate and find critical points ($f'(x) = 0$):
Compute first derivative and solve for critical variable value. [1.0 Mark]
Step 4: Execute Second Derivative Test ($f''(x)$):
Verify sign of second derivative to confirm maximum/minimum. [1.0 Mark]
Step 5: Calculate optimal dimensions/value:
Substitute critical value back to state complete dimensions. [1.0 Mark]
Conclusion: Optimal dimensions derived successfully.

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🎯 Official CBSE Marking Scheme:
Setting up constraint and objective function: 1.0 Mark
Evaluating first derivative f'(x): 1.0 Mark
Finding critical points: 1.0 Mark
Second derivative test for max/min: 1.0 Mark
Evaluating final dimensions/maximum value: 1.0 Mark