Question

If function $f\left(x\right)=\{\begin{array}{cc}1,& x=1\\ a^2-3a+x^2,& x>1\end{array}$ has a local maximum at $x=1,$ then the set of values of $a$ is

• A. $(3,\infty )$
• B. $(-\infty ,0)$
• C. $(-\infty ,3)$
• D. $(0,3)$

Answer 1

The correct option is D $(0,3)$

For maximum at $x=1:$
$f\left(1\right)>f(1+h)$ as $h\to 0^{+}$
$\Rightarrow 1>a^2-3a+(1+h{)}^2$ as $h\to 0^{+}$
$\Rightarrow a^2-3a<0$
$\Rightarrow a\in (0,3)$