Differential Equations 1 Question 23
23. Consider the statements.
I. There exists some $x \in R$ such that, $f(x)+2 x=2\left(1+x^{2}\right)$
II. There exists some $x \in R$ such that,
$2 f(x)+1=2 x(1+x)$
(a) Both I and II are true
(b) I is true and II is false
(c) I is false and II is true
(d) Both I and II are false
Show Answer
Solution:
- Here, $f(x)+2 x=(1-x)^{2} \cdot \sin ^{2} x+x^{2}+2 x$
where, I: $f(x)+2 x=2(1+x)^{2}$
$$ \begin{aligned} & \therefore \quad 2\left(1+x^{2}\right)=(1-x)^{2} \sin ^{2} x+x^{2}+2 x \\ & \Rightarrow \quad(1-x)^{2} \sin ^{2} x=x^{2}-2 x+2 \\ & \Rightarrow \quad(1-x)^{2} \sin ^{2} x=(1-x)^{2}+1 \\ & \Rightarrow \quad(1-x)^{2} \cos ^{2} x=-1 \end{aligned} $$
which is never possible.
$\therefore I$ is false.
Again, let $h(x)=2 f(x)+1-2 x(1+x)$
where,
$$ \begin{aligned} & h(0)=2 f(0)+1-0=1 \\ & h(1)=2(1)+1-4=-3 \text { as }[h(0) h(1)<0] \end{aligned} $$
$\Rightarrow h(x)$ must have a solution.
$\therefore$ II is true.
