Question

The number of integral value(s) of $a$ for which $f\left(x\right)=\{\begin{array}{cc}{a}^2-3+\sin x,& 0<x<\frac{\pi }{2}\\ 1+\cos x,& \frac{\pi }{2}\le x<\pi \end{array}$ has a point of local maxima at $x=\frac{\pi }{2},$ is

Answer 1

Given : $f\left(x\right)=\{\begin{array}{cc}{a}^2-3+\sin x,& 0<x<\frac{\pi }{2}\\ 1+\cos x,& \frac{\pi }{2}\le x<\pi \end{array}$, has a point of local maxima at $x=\frac{\pi }{2}$
$\Rightarrow f\left({\frac{\pi }{2}}^{-}\right)<f\left(\frac{\pi }{2}\right)>f\left({\frac{\pi }{2}}^{+}\right)$
$\Rightarrow a^2-3+1<1+0$ and $\cos x$ is decreasing in $[\frac{\pi }{2},\pi ],$ so $f\left(\frac{\pi }{2}\right)>f\left({\frac{\pi }{2}}^{+}\right)$
$\Rightarrow a^2-3<0$
$\Rightarrow a\in (-\sqrt{3},\sqrt{3})$
So, possible integral values of $a$ is $-1,0,1$