Question

Let $f\left(x\right)=\{\begin{array}{cc}{x}^3-x^2+10x-7,& x\le 1\\ -2x+{\log }_2(k^2-4),& x>1\end{array}$
The set of values of $k$ for which $f\left(x\right)$ has greatest value at $x=1$, is

• A. $[-6,6]$
• B. $[-6,-2]\cup [2,6]$
• C. $[-6,-2)\cup (2,6]$
• D. $(2,6)$

Answer 1

The correct option is C $[-6,-2)\cup (2,6]$

As $f\left(x\right)$ has maxima at $x=1$, all values of $f\left(x\right)$ in the right neighbourhood of $x=1$ will be less than or equal to $f\left(1\right)$

$\underset{h\to 0}{lim}f(1+h)\le f\left(1\right)$
$-2+{\log }_2(k^2-4)\le 1-1+10-7$
$k^2-4\le 2^5$ and $k^2-4>0$
$k^2\le 36$ and $k^2>4$
$k\in [-6,-2)\cup (2,6]$