$f\left(x\right)=(x-8{)}^4(x-9{)}^5,0\le x\le 10$ , monotonically decreases in

$f\left(x\right)=(x-8{)}^4(x-9{)}^5,0\le x\le 10$

⇒f′(x)=$$\begin{gathered} 4{(x-8)}^3{(x-9)}^5+5{(x-9)}^4{(x-8)}^4 \\ ={(x-8)}^3{(x-9)}^4\left(4\right(x-9)+5(x-8\left)\right) \end{gathered}$$
 

$\therefore$ $f\text{'}\left(x\right)=9(x-8{)}^3(x-9{)}^4(x-\frac{76}{9})$

sign scheme of f '(x) is as follows;

$f^{\mathrm{\prime }}\left(x\right)<0 \text{if }x\in (8,\frac{76}{9})$ , i.e., $f\left(x\right)$  decreases if  $x\in (8,\frac{76}{9})$.