The number of solutions of the equation ^-1[x^2+13]+^-1[x^2-23]=x^2, for x[-1,1], where [x…

Question

The number of solutions of the equation $\sin^{-1}\left[x^2+\dfrac13\right]+\cos^{-1}\left[x^2-\dfrac23\right]=x^2$, for $x\in[-1,1]$, where $[x]$ denotes the greatest integer function, is

Accepted Answer

Writing $\left[x^2+\frac13ight]=\left[x^2-\frac23ight]+1$ and using $[t+n]=[t]+n$ for integer $n$, both bracket terms depend on $\left[x^2-\dfrac23ight]$, which (since $0\le x^2\le1$) can only equal $-1$ or $0$.

Case $\left[x^2-\frac23ight]=0$: equation becomes $\sin^{-1}(1)+\cos^{-1}(0)=x^2\Rightarrow x^2=\pi$ --- but then $x^2-\dfrac23=\pi-\dfrac23
e0$, a contradiction.

Case $\left[x^2-\frac23ight]=-1$: equation becomes $\sin^{-1}(0)+\cos^{-1}(-1)=x^2\Rightarrow x^2=\pi$ --- again contradicts $x^2-\dfrac23=-1$.

Both cases fail, so there are 0 solutions.

Solutions · 1