Integral Calculus Question 172
Question: The line $ y=\alpha $ intersects the curve $ y=g(x) $ ,atleast at two points. If $ \int\limits_2^{x}{g(t)dt=\frac{x^{2}}{2}+\int\limits_x^{2}{t^{2}g(t)dt}} $ then possible value of $ \alpha $ is/are-
Options:
A) $ ( -\frac{1}{2},\frac{1}{2} ) $
B) $ [ -\frac{1}{2},\frac{1}{2} ] $
C) $ ( -\frac{1}{2},\frac{1}{2} )-\{0\} $
D) $ \{ -\frac{1}{2},0,\frac{1}{2} \} $
Show Answer
Answer:
Correct Answer: C
Solution:
[c] $ \int\limits_2^{x}{g(t)dt=\frac{x^{2}}{2}+\int\limits_x^{2}{t^{2}g(t)dt}} $ Differentiating w.r.t. x, we get $ g(x)=x+(-x^{2}(g(x))\Rightarrow g(x)=\frac{x}{1+x^{2}} $ Clearly from graph, $ \alpha \in ( -\frac{1}{2},\frac{1}{2} )-{0} $
BETA
