IIM B UG (B.Sc-Hons.)- Sample Paper 2026 · Quant & DI · Question 1
IIM B UG (B.Sc-Hons.)- Sample Paper 2026 question on Application of Derivatives, with the answer and a step-by-step solution.
A rectangle has its lower vertices on the x-axis and its upper vertices on the parabola y = 12 -{x}^{2}. Its sides are parallel to the coordinate axes, and it lies above the x-axis. If its area is as large as possible, what is the length of its diagonal?
- A2\sqrt{13} units
- B8 units
- C4\sqrt{5} units
- D4\sqrt{6} units
Pick an option, then check.
Answer & solution
Answer: Option C
By symmetry the upper vertices are at (−x, 12 − x²) and (x, 12 − x²), so the width is 2x and the height is 12 − x².
Area A = 2x(12 − x²) = 24x − 2x³.
Differentiate and set to zero: dA/dx = 24 − 6x² = 0, so x² = 4 and x = 2.
The second derivative is −12x, which is negative at x = 2, so this is a maximum.
Width = 2x = 4 and height = 12 − 4 = 8.
Diagonal = √(4² + 8²) = √80 = 4√5 units.
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