Question

Let a function $f\left(x\right)=x+\cos x$ and $f\left(x\right)=0$ has atleast one real solution, then which of the following is correct

• A. $x\in (0,1)$
• B. $x\in (1,2)$
• C. $x\in (-1,1)$
• D. $x\in (-1,0)$

Answer 1

Answer: (C), (D)

$x\in (-1,0)$

Given : $f\left(x\right)=x+\cos x$ is continuous and monotonic in $R.$
$\Rightarrow I.V.T.$ is applicable in any interval.
Now, $f\left(0\right)=1>0,f\left(1\right)=1+\cos 1>0$
$\Rightarrow$ no solution.
$f(-1)=-1+\cos 1<0,f\left(1\right)=1+\cos 1>0$
$f\left(2\right)=2+\cos 2>0$
$\Rightarrow f\left(x\right)=0$ has atleast one solution in $(-1,1)$ or $(-1,0)$ but no solution in $(0,1)$ or $(1,2)$