Integral-Calculus Question 523
Question: If $ \int\limits_1^{2}{{ K^{2}+(4-4K)x+4x^{3} }dx\le 12} $ , then which one of the following is correct?
Options:
A) $ K=3 $
B) $ 0\le K<3 $
C) $ K\le 4 $
D) $ K=0 $
Show Answer
Answer:
Correct Answer: A
Solution:
[a] Let $ \int\limits_1^{2}{{ K^{2}+( 4-4K )x+4x^{3} }dx\le 12} $
$ \Rightarrow K^{2}x+. \frac{(4-4K)x^{2}}{2}+\frac{4x^{4}}{4} |_1^{2}\le 12 $
$ \Rightarrow [2K^{2}+(2-2K)(4)+16]-[K^{2}+(2-2K) $ $ +1]\le 12 $
$ \Rightarrow (2K^{2}+8-8K+16)-(K^{2}-2K+3)\le 12 $
$ \Rightarrow K^{2}-6K+21\le 12 $
$ \Rightarrow K^{2}-6K+9\le 0\Rightarrow {{(K-3)}^{2}}\le 0 $
$ \Rightarrow K=3 $
BETA
