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Application of Derivatives
NCERT Class 12
CBSE
Grade 12
Question:
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
Step-by-Step Solution
Express Area $A(\theta) = 2R^2 \sin 2\theta$. [1.5 Marks] $\dfrac{dA}{d\theta} = 4R^2 \cos 2\theta = 0 \Rightarrow 2\theta = \pi/2 \Rightarrow \theta = \pi/4$. [1.5 Marks] Verify $\dfrac{d^2 A}{d\theta^2} = -8R^2 < 0 \Rightarrow$ Maximum area. [0.5 Mark] At $\theta = \pi/4$: $x = y = \sqrt{2} R \Rightarrow$ Rectangle is a square. Proved! [1.5 Marks]
--- 🎯 Official CBSE Marking Scheme: Setting up area function using parametric angle $\theta$: 1.5 Marks Evaluating critical angle $\theta = \pi/4$: 1.5 Marks Applying Second Derivative Test: 0.5 Mark Proving $x = y = \sqrt{2} R$ (Square): 1.5 Marks
Correct Answer:
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