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

Step 1: Solving the integral.
The integral of \( x \) from 0 to \( a \) is: \[ \int_0^a x \, dx = \frac{a^2}{2} \] Thus, the inequality becomes: \[ \frac{a^2}{2} \leq \frac{a}{2} + 6 \] Multiply both sides by 2 to simplify: \[ a^2 \leq a + 12 \] Rearrange the terms: \[ a^2 - a - 12 \leq 0 \] Step 2: Solving the quadratic inequality.
Factor the quadratic expression: \[ (a - 4)(a + 3) \leq 0 \] The solution to this inequality is \( -3 \leq a \leq 4 \).