let ( f(x)=\frac{x}{x + 7} ). find the values of ( x ) where ( f^{prime}(x)=6 ).\ngive exact answers (not…

Explanation:

Step1: Differentiate ( f(x) ) using the quotient rule

The quotient rule states that if ( f(x)=\frac{u}{v} ), then ( f^{\prime}(x)=\frac{u^{\prime}v - uv^{\prime}}{v^{2}} ). Here, ( u = x ), ( u^{\prime}=1 ), ( v=x + 7 ), ( v^{\prime}=1 ). So ( f^{\prime}(x)=\frac{1\cdot(x + 7)-x\cdot1}{(x + 7)^{2}}=\frac{x + 7-x}{(x + 7)^{2}}=\frac{7}{(x + 7)^{2}} ).

Step2: Set ( f^{\prime}(x)=6 ) and solve for ( x )

Set ( \frac{7}{(x + 7)^{2}}=6 ). Cross - multiply to get ( 7 = 6(x + 7)^{2} ). Then ( (x + 7)^{2}=\frac{7}{6} ). Take square roots: ( x+7=\pm\sqrt{\frac{7}{6}}=\pm\frac{\sqrt{42}}{6} ). Solve for ( x ): ( x=-7\pm\frac{\sqrt{42}}{6} ).

Answer:

The greater solution is ( x=-7+\frac{\sqrt{42}}{6} ). The lesser solution is ( x=-7-\frac{\sqrt{42}}{6} ).