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

$$\begin{gathered} f\left(1\right)=3 \\ \text{ For }\mathrm{x}<1,\mathrm{f}\text{'}\left(x\right)=3{\mathrm{x}}^2-2\mathrm{x}+10>0 \Rightarrow \mathrm{f}\left(\mathrm{x}\right)\text{ is increasing } \\ \mathrm{For} x>1,f\text{'}\left(x\right)<0\Rightarrow \mathrm{f}\left(\mathrm{x}\right)\text{ is decreasing } \\ \underset{x\to 1^{+}}{lim} f\left(x\right)=-2+{\log }_2\left(b^2-4\right) \\ \text{ For maximum value at }\mathrm{x}=1 \end{gathered}$$

$\begin{array}{l}\Rightarrow 3\ge -2+{\log }_2\left(b^2-4\right)\\ {\log }_2\left(b^2-4\right)\le 5\\ b^2-4>0 and b^2-4\le 32 or b^2-36\le 0\\ \boxed{\mathrm{b}\in [-6,-2)\cup (2,6]}\end{array}$