The maximum area (in sq. units) of a rectangle having its base on the x‑axis and its other two vertices on the parabola, y = 12 – x2 such that the rectangle lies inside the parabola, is

Engineering

Mathematics

Maxima and Minima

maximum and minimum of quadratic and rational expression

Question

The maximum area (in sq. units) of a rectangle having its base on the x‑axis and its other two vertices on the parabola, y = 12 – x² such that the rectangle lies inside the parabola, is

$18 \sqrt{3}$

$20 \sqrt{2}$

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A = 2x · (12 – x²)

A = 24x – 2x³ ; $x \in (0, 2 \sqrt{3})$

$\frac{dA}{dx} = 24 - 6 x^{2} = 6 (4 - x^{2})$

= – 6 (x – 2) (x + 2)

$∴$ A is max at x = 2

$∴$ A = 2 · 2 · (12 – 4) = 4 · 8 = 32