If \( \int_0^a x \, dx \leq \frac{a}{2} + 6 \), then which of the following holds for \( a \)?
Show Hint
- \( -4 \leq a \leq 3 \)
- \( a \geq 4, a \leq -3 \)
- \( -3 \leq a \leq 4 \)
- \( -3 \leq a \leq 0 \)
The Correct Option is C
Solution and Explanation
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.
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