Question

All the values of '$a$' for which ${\int }_1^2\{a^2+(4-4a)x+4x^3\}dx\le 12$ are given by

• A. $a=3$
• B. $a\le 4$
• C. $0\le a<3$
• D. none of these

Answer 1

The correct option is A $a=3$

$I={\int }_1^2\{a^2+(4-4a)x+4x^3\}dx\le 12\Rightarrow [a^{2}x+(2-2a)x^2+x^4{]}_1^2\le 12\Rightarrow a^2+(2-2a\left)\right(4-1)+(16-1)\le 12\Rightarrow a^2+6-6a+15\le 12\Rightarrow a^2-6a+9\le 0\Rightarrow (a-3{)}^2\le 0\Rightarrow a=3$