Question

If \( \int_0^a x \, dx \leq \frac{a}{2} + 6 \), then which of the following holds for \( a \)?

• A. \( -4 \leq a \leq 3 \)
• B. \( a \geq 4, a \leq -3 \)
• C. \( -3 \leq a \leq 4 \)
• D. \( -3 \leq a \leq 0 \)

Hint:

For quadratic inequalities, factor the expression and analyze the sign of the factors to determine the solution range.

Answer 1

The Correct Option is C

To address the problem, we must first identify the interval of \( a \) values that fulfill the given inequality:

\( \int_0^a x \, dx \leq \frac{a}{2} + 6 \)

1. Integration Calculation:
Compute the definite integral on the left side.

\( \int_0^a x \, dx = \left[ \frac{x^2}{2} \right]_0^a = \frac{a^2}{2} \)

2. Inequality Formulation:
Substitute the integral result into the inequality.

\( \frac{a^2}{2} \leq \frac{a}{2} + 6 \)

3. Denominator Elimination:
Multiply the inequality by 2 to remove fractions.

\( a^2 \leq a + 12 \)

4. Inequality Rearrangement:
Rewrite the inequality to one side.

\( a^2 - a - 12 \leq 0 \)

5. Quadratic Inequality Solution:
Factor the quadratic expression.
\( (a - 4)(a + 3) \leq 0 \)
This inequality holds true for the following range:
\( -3 \leq a \leq 4 \)

6. Determination of the Range:
The definitive range for \( a \) is determined to be \( -3 \leq a \leq 4 \)

Final Answer:
The correct option is (C) -3 ≤ a ≤ 4.