Find an equation of the tangent line to the graph of the function $f ( x ) = x + \frac { 1 } { x }$ at the point that has x-coordinate $x = \frac { 1 } { 4 }$ .
A) $y = - 15 x + 8$
B) $y = - 15 x - 120$
C) $y = 8 x - 15$
D) $y = x - 15$
E) $y = 15 x + 8$

Question

Find an equation of the tangent line to the graph of the function $f ( x ) = x + \frac { 1 } { x }$ at the point that has x-coordinate $x = \frac { 1 } { 4 }$ .  
A) $y = - 15 x + 8$
B) $y = - 15 x - 120$
C) $y = 8 x - 15$
D) $y = x - 15$
E) $y = 15 x + 8$

Quizplus · Accepted Answer

To find the equation of the tangent line to the graph of $f(x) = x + \frac{1}{x}$ at $x = \frac{1}{4}$, we first find the derivative $f'(x)$ to get the slope of the tangent line at that point. The derivative is $f'(x) = 1 - \frac{1}{x^2}$. Substituting $x = \frac{1}{4}$ into $f'(x)$, we get $f'(\frac{1}{4}) = 1 - 16 = -15$, which is the slope of the tangent line. To find the y-coordinate of the point of tangency, we substitute $x = \frac{1}{4}$ into $f(x)$, yielding $f(\frac{1}{4}) = \frac{1}{4} + 4 = \frac{17}{4}$. The point of tangency is $(\frac{1}{4}, \frac{17}{4})$. Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, with $m = -15$ and $(x_1, y_1) = (\frac{1}{4}, \frac{17}{4})$, we get $y - \frac{17}{4} = -15(x - \frac{1}{4})$. Simplifying, we find $y = -15x + 8$, which matches choice A.

Find an Equation of the Tangent Line to the Graph f(x)=x+1xf ( x ) = x + \frac { 1 } { x }f(x)=x+x1​

Find an Equation of the Tangent Line to the Graph $f ( x ) = x + \frac { 1 } { x }$