Question

Let a function $f\left(x\right)=\{\begin{array}{cc}{a}^2-2a-3+5x^3,& x<0\\ -2x^2,& x\ge 0\end{array}$ has a local maximum at $x=0.$ Then set of values of $a$ is

• A. $(-\infty ,-1]$
• B. $[-1,3]$
• C. $(-\infty ,3]$
• D. $[3,\infty )$

Answer 1

The correct option is B $[-1,3]$

$f\left(x\right)=\{\begin{array}{cc}{a}^2-2a-3+5x^3,& x<0\\ -2x^2,& x\ge 0\end{array}$
Plotting the graph of $f\left(x\right):$

Here, $k=a^2-2a-3.$
Clearly from graph : $k\le 0$ for maximum at $x=0$
$\Rightarrow a^2-2a-3\le 0$
$\therefore a\in [-1,3]$