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 solve the problem, we need to determine the range of values of \( a \) that satisfy the inequality:

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

1. Evaluate the Definite Integral:
We evaluate the integral on the left-hand side:

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

2. Set Up the Inequality:
Now substitute into the inequality:

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

3. Multiply Through by 2 to Eliminate Denominators:
\( a^2 \leq a + 12 \)

4. Rearrange the Inequality:
\( a^2 - a - 12 \leq 0 \)

5. Solve the Quadratic Inequality:
Factor the quadratic:
\( (a - 4)(a + 3) \leq 0 \)
This inequality is satisfied when:
\( -3 \leq a \leq 4 \)

6. Conclusion:
The correct range of values for \( a \) is \( -3 \leq a \leq 4 \)

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