Question

Can't figure it out!!

Answer 1

$\bf slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ f(x_2)}}-{{ f(x_1)}}}{{{ x_2}}-{{ x_1}}}\impliedby \begin{array}{llll} average\ rate\\ of\ change \end{array}\\\\ -------------------------------\\\\ f(x)= \cfrac{1}{x} \qquad \begin{cases} x_1=2\\ x_2=b \end{cases}\implies \cfrac{f(b)-f(2)}{b-2}=-\cfrac{1}{10} \\\\\\ \cfrac{\frac{1}{b}-\frac{1}{2}}{b-2}=-\cfrac{1}{10}\implies \cfrac{\frac{2-b}{2b}}{b-2}=-\cfrac{1}{10}\implies \cfrac{2-b}{2b}=-\cfrac{b-2}{10}$

$\bf \cfrac{2-b}{2b}=\cfrac{2-b}{10}\implies 20-10b=4b-2b^2\impliedby cross-multiplying \\\\\\ 2b^2-4b-10b+20=0\implies 2b^2-14b+20=0 \\\\\\ b^2-7b+10=0\implies (b-2)(b-5)=0\implies b= \begin{cases} \boxed{5}\\ 2 \end{cases}$