Let $f(x)=\left\{\begin{matrix}x^3+x^2-10x,&-1≤x<0\\\cos x,&0≤x<\frac{π}{2}\\1+\sin x,&\frac{π}{2}≤x<π\end{matrix}\right\}$. Then at $x=\frac{π}{2}$, f(x) has:

A local maximum

$f(x)=\left\{\begin{matrix}x^3+x^2-10x,&-1≤x<0\\\cos x,&0≤x<\frac{π}{2}\\1+\sin x,&\frac{π}{2}≤x<π\end{matrix}\right\}$

We have to find the condition at $x=\frac{π}{2}$

So just consider last 2 functions; So f(x) has a local maximum