Mathematics - Application of Derivative Question with Solution | TestHub

Given, $f\left(x\right)=\left\{\begin{array}{cc}{x}^3-x^2+10x-7,& x\le 1\\ -2x+{\log }_2\left(b^2-4\right),& x>1\end{array}\right.$

Now $f\left(1\right)=3$

For $x<1,f^{\text{'}}\left(x\right)=3x^2-2x+10>0$

$\Rightarrow f\left(x\right)$ is increasing

For $x>1,f^{\text{'}}\left(x\right)<0$

$\Rightarrow$  function is decreasing.

So, at $x=1$ there is a point of maxima

So, $\underset{x\to 1^{+}}{\lim }f\left(x\right)=-2+{\log }_2\left(b^2-4\right)$

For maximum value at $x=1$

$\Rightarrow 3\ge -2+{\log }_2\left( {\mathrm{b}}^2-4\right)$

$\Rightarrow 32\ge b^2-4>0$

$\Rightarrow b\in [-6,-2)\cup (2,6]$