Question

Multiple Choice Which of the following is an equation of the normal to the graph of \(f(x)=2 / x\) at \(x=1 ? \quad\) $$\begin{array}{ll}{\text { (A) } y=\frac{1}{2} x+\frac{3}{2}} & {\left(\text { B ) } y=-\frac{1}{2} x \quad \text { (C) } y=\frac{1}{2} x+2\right.} \\\ {\text { (D) } y=-\frac{1}{2} x+2} & {\text { (E) } y=2 x+5}\end{array}$$

Short answer

The equation of the normal line to \(f(x)=2 / x\) at \(x=1\) is \(y = 1/2x + 3/2\), which corresponds to option (A).

Step-by-step solution

Find the derivative

The first step is to find the derivative of the function. Using the power rule for differentiation, the derivative of \(f(x)=2 / x\) is \(f'(x)=-2 / x^2\).

Evaluate the derivative at the given point

Next, we plug the given x-value from the question (which is 1) into the derivative to find the slope of the tangent line. So, \(f'(1)=-2 / (1)^2 = -2\).

Find the negative reciprocal of the slope

The normal line is perpendicular to the tangent line, which means its slope is the negative reciprocal of the slope of the tangent line. The negative reciprocal of -2 is \(1/2\). So the slope of the normal line is \(1/2\).

Find the equation of the normal line

At x=1, \(f(x)=2\), so the point on the graph where our normal line will cross is (1,2). With this point and our slope, we can use the point-slope formula \(y - y1 = m(x - x1)\) to find our equation: \(y - 2 = 1/2(x - 1)\), which simplifies to \(y = 1/2x + 3/2\).

Compare to multiple-choice options

The last step is to compare our equation to the multiple-choice options in the question. In this case, we see that our equation matches option (A).