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A polynomial $P$ is positive for $x > 0,$ and the area of the region bounded by $P (x),$ the x-axis, and the vertical lines $x = 0$ and $x = K$ is $K^{2} (K + 3) / 3$ .The polynomial $P (x)$ is

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$x^{2} + 2 x$

$x^{2} + x + 1$

$x^{2} + 2 x + 1$

$x^{3} + 1$

answer is A.

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Detailed Solution

We have $\int_{0}^{K} P (t) d t = \frac{K^{2} (K + 3)}{3}$ Differentiating

$w . r . t K,$ we get $P (K) = \frac{K^{2}}{3} + \frac{2 K}{3} (K + 3) = K^{2} + 2 K$ .

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