Question

Let $f\left(x\right)=\{\begin{array}{cc}x,& 0\le x\le 1\\ -x^2+k,& 1<x\le 2\end{array}$ has a local maximum at $x=1.$ Then the number of non-negative integral value(s) of $k$ is

Answer 1

Given : $f\left(x\right)=\{\begin{array}{cc}x,& 0\le x\le 1\\ -x^2+k,& 1<x\le 2\end{array}$
For maximum at $x=1:$
$f(1-h)<f\left(1\right)>f(1+h)$ as $h\to 0^{+}$
$\Rightarrow 1-h<1$ and $-(1+h{)}^2+k<1$ as $h\to 0^{+}$
$\Rightarrow -h<0$ and $k<1+(1+h^2+2h)$ as $h\to 0^{+}$
$\Rightarrow -h<0$ and $k<2+h^2+2h$ as $h\to 0^{+}$
$\Rightarrow k\in R$ and $k\le 2$
$\therefore k\le 2$
So, total non-negative integral values of $k$ are $3.$