The slope of a function $$y = x^3 + kx  at  x = 2$$ is equal to the area under the curve $$z = a^2 + a$$ between points a = 0 and a = 3 Then the value of k is

The slope of the function $$y = x^3 + kx$$  at x = 2 can be calculated as shown below,

Slope = $$\dfrac{dy}{dx}\ =\ 3x^2\ +\ k$$

Slope at x = 2 is $$\ 3\left(2\right)^2\ +\ k\ =\ 12\ +\ k$$

The area under the curve $$z = a^2 + a$$ between points a = 0 and a = 3 can be calculated as,

Area = $$\int\ _0^3\left(a^2\ +\ a\right)da\ =\ \left[\dfrac{a^3}{3}\ +\ \dfrac{a^2}{2}\right]_{_0}^{^{^3}}\ =\ \dfrac{3^3}{3}\ +\ \dfrac{3^2}{2}\ -\ 0\ =\ 9\ +\ 4.5\ =\ 13.5$$

Given that the area is equal to the slope. Equating them, we get,

13.5 = 12 + k

k = 1.5

Hence, the correct answer is option A.