Question

If: $[-20,20]\to R$ defined by $f\left(x\right)=\left[\frac{x^2}{a}\right]\sin x+\cos x$ is an even function, then the set of value of 'a'. (where $[\cdot ]$ denotes the greatest integral function.) is $aϵ(5b,\infty )$ Find b

Answer 1

For given function to be even, $\left[\frac{x^2}{a}\right]=0\forall x\in [-20,20]$
$\Rightarrow 0\le \frac{x^2}{a}<1\Rightarrow a>x^2\forall x\in [-20,20]\Rightarrow a>{20}^2=400$
$\therefore a\in (40,\infty )$. Hence $b=8$