Find two positive numbers $x$ and $y$ such that $x + y = 60$ and $x y^3$ is maximum.
Step-by-Step Solution
Given: Problem statement: Find two positive numbers $x$ and $y$ such that $x + y = 60$ and $x y^3$ is maximum.
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.
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🎯 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: