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(D) Let $g(x) = \frac{ax^3}{3} + \frac{bx^2}{2} + cx.$Then $g'(x) = ax^2 + bx + c.$We are given $2a + 3b + 6c = 0.$Statement $2:$$g(0) = 0.$$g(1) = \f

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Statement $1$ is true,Statement $2$ is true,Statement $2$ is a correct explanation for Statement $1.$

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(D) Let $g(x) = \frac{ax^3}{3} + \frac{bx^2}{2} + cx.$Then $g'(x) = ax^2 + bx + c.$We are given $2a + 3b + 6c = 0.$Statement $2:$$g(0) = 0.$$g(1) = \frac{a}{3} + \frac{b}{2} + c = \frac{2a + 3b + 6c}{6} = \frac{0}{6} = 0.$Since $g(0) = g(1) = 0$ and $g(x)$ is a polynomial,it is continuous on $[0, 1]$ and differentiable on $(0, 1).$By Rolle's theorem,there exists at least one $k \in (0, 1)$ such that $g'(k) = 0.$Thus,Statement $2$ is true.Statement $1:$Since $g'(k) = ak^2 + bk + c = 0$ for some $k \in (0, 1),$ the quadratic equation $ax^2 + bx + c = 0$ has at least one root in $(0, 1).$Thus,Statement $1$ is true and Statement $2$ is the correct explanation for Statement $1.$

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