To solve this problem, we start by evaluating the given integral:
x‌dxThe integral of
x with respect to
x from 0 to
a can be calculated as follows:
x‌dx=[‌]0a=‌−‌=‌ We are given that:
x‌dx≤a+4Substituting the calculated integral, we get:
‌≤a+4To solve this inequality, multiply each term by 2 to clear the fraction:
a2≤2a+8 Rewrite it as a standard quadratic inequality:
a2−2a−8≤0We factorize the quadratic expression:
a2−2a−8=(a−4)(a+2)Thus, the inequality becomes:
(a−4)(a+2)≤0 To solve this inequality, we find the critical points where the expression equals zero, which are
a=4 and
a=−2. Test the intervals determined by these points to see where the inequality holds true:
1. For
a<−2, choose
a=−3 :
(−3−4)(−3+2)=(−7)(−1)=7>02. For
−2≤a≤4, choose
a=0 :
(0−4)(0+2)=(−4)(2)=−8≤03. For
a>4, choose
a=5 :
(5−4)(5+2)=(1)(7)=7>0 The inequality
(a−4)(a+2)≤0 holds true for:
−2≤a≤4Therefore, the correct answer is:
Option C
−2≤a≤4