find the values of ( x ) where the tangent line to the graph of ( f(x)=\frac{1}{x} ) is parallel to the line…

find the values of ( x ) where the tangent line to the graph of ( f(x)=\frac{1}{x} ) is parallel to the line ( y=-3 x + 4 ). enter the exact values of the answer(s) (not decimal approximations).( x = )

find the values of ( x ) where the tangent line to the graph of ( f(x)=\frac{1}{x} ) is parallel to the line ( y=-3 x + 4 ). enter the exact values of the answer(s) (not decimal approximations).( x = )

Answer

Explanation:

Step1: Find the derivative of ( f(x) )

Using the power rule ( (x^n)^\prime=nx^{n - 1} ), for ( f(x)=\frac{1}{x}=x^{-1} ), then ( f^\prime(x)=-1\times x^{-2}=-\frac{1}{x^{2}} ).

Step2: Determine the slope of the given line

The line ( y = - 3x+4 ) is in the form ( y = mx + b ) (slope - intercept form), where ( m=-3 ).

Step3: Set the derivative equal to the slope of the line

Since the tangent line is parallel to ( y=-3x + 4 ), their slopes are equal. So we set ( f^\prime(x)=-3 ), i.e., ( -\frac{1}{x^{2}}=-3 ). Multiply both sides by ( -x^{2} ): ( 1 = 3x^{2} ). Then ( x^{2}=\frac{1}{3} ). Take the square root of both sides: ( x=\pm\frac{1}{\sqrt{3}}=\pm\frac{\sqrt{3}}{3} ).

Answer:

(x = \frac{\sqrt{3}}{3},x=-\frac{\sqrt{3}}{3})