For a real number $a$, let $I(a) = \int_{-1}^1 (3x^2 - ax + 1) \, dx$. Which of the following statements is/are true?

To evaluate the integral $I(a)$:

Using the linearity of integration and properties of even and odd functions:

• $3x^2 + 1$ is an even function, so $\int_{-1}^1 (3x^2 + 1) \, dx = 2 \int_0^1 (3x^2 + 1) \, dx = 2 \left[ x^3 + x \right]_0^1 = 2(1 + 1) = 4$.

• $-ax$ is an odd function, so $\int_{-1}^1 (-ax) \, dx = 0$ for any real constant $a$.

Alternatively, integrating directly:

Since $I(a) = 4$ for all $a \in (-\infty, +\infty)$:

• (A) is true because $I(a)$ is a constant ($4$) and does not depend on $a$.

• (B) is false because the value cannot vary with $a$.

• (C) is true because $I(a) = 4$, which is a positive real number.

• (D) is false because $I(a)$ is never negative for any real $a$.

Hence, statements A and C are correct.