Question

Let $f(x)= \begin{cases} x^3-x^2+10 x-7, & x \leq 1 \\ -2 x+\log _2\left(b^2-4\right), & x>1\end{cases}$
Then the set of all values of $b$, for which $f(x)$ has maximum value at $x=1$, is :

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

Answer 1

Answer: (C)

$ f (1)=3 $
For $ x < 1, f ^{\prime}( x )=3 x ^2-2 x +10>0$
$ \Rightarrow f ( x ) $ is increasing
For $ x >1, f ^{\prime}( x ) < 0 $
$ \Rightarrow $ function is decreasing.
$\displaystyle\lim _{ x \rightarrow 1^{+}} f ( x )=-2+\log _2\left( b ^2-4\right) $
For maximum value at $ x =1$
$ 3 \geq-2+\log _2\left( b ^2-4\right) $
$ 32 \geq b ^2-4>0$
$ b \in[-6,-2) \cup(2,6]$