GATE CS 2026 Set 2 Q17 — Maxima and Minima, Mean Value Theorem, Integration

To determine the correct statements, let us evaluate the definite integral $I(a)$:

Using the linearity property of integration, we can split the integral into three parts:

Now, we evaluate each term individually:

1. First term: $$\int_{-1}^{1} 3x^2 \, dx = \left[ x^3 \right]_{-1}^{1} = 1^3 - (-1)^3 = 1 - (-1) = 2$$
2. Second term: Since $x$ is an odd function and the integration interval $[-1, 1]$ is symmetric about $0$: $$\int_{-1}^{1} x \, dx = 0 \implies a \int_{-1}^{1} x \, dx = 0$$
3. Third term: $$\int_{-1}^{1} 1 \, dx = \left[ x \right]_{-1}^{1} = 1 - (-1) = 2$$

Substituting these values back into the expression for $I(a)$:

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

• Option A is True: The value of $I(a)$ is constant ($4$) and does not depend on $a$.
• Option B is False: The value of $I(a)$ is independent of $a$ and does not vary.
• Option C is True: Since $I(a) = 4 > 0$ for all real $a$, there exists $a$ (in fact, any real $a$) such that $I(a)$ is a positive real number.
• Option D is False: $I(a)$ is always $4$, so it can never be negative.