Question

Horizontal Tangent Line Determine the point(s) at which the graph of $$f(x)=\frac{x}{\sqrt{2 x-1}}$$ has a horizontal tangent.

Short answer

The point(s) at which the graph of the function \( f(x)=\frac{x}{\sqrt{2 x-1}} \) has a horizontal tangent can be obtained by finding the \( x \)-values for which the derivative of this function equals zero. Then, plugging these \( x \)-values back into the function yields the respective \( y \)-coordinates.

Step-by-step solution

Derive the Function

Begin by finding the derivative of the function. Use the quotient rule, which states that if \( y = \frac{u}{v} \), then \( y' = \frac{v.u'-u.v'}{v^{2}} \). Here, \( u = x \) and \( v = \sqrt{2 x-1} \). Hence, \(u'=1\) and \(v'=\frac{1}{\sqrt{2 x-1}}\). Apply the quotient rule and simplify the derivative.

Set the Derivative Equal to Zero

A horizontal tangent is characterized by a derivative of zero. Hence, set \( f'(x) \) equal to zero and solve for \( x \).

Find the x-coordinates of the Points

The solution(s) to the equation \( f'(x) = 0 \) will give the \( x \)-coordinates of the point(s) where the graph has a horizontal tangent.

Find the Corresponding y-coordinates

Substitute the \( x \)-values obtained from Step 3 into the original function \( f(x) \) to find the corresponding \( y \)-coordinates. This will yield the point(s) at which the graph has a horizontal tangent.

Key concepts

Derivative of a Function

Understanding the derivative of a function is essential in calculus, as it measures how a function changes as its input changes. It is the mathematical equivalent of determining the slope of the tangent line at any point on a function's graph. The derivative of a function at a particular point can be visually understood as the slope of the line that just barely touches the function at that point.

When you calculate the derivative of a function, you're looking for a new function, often referred to as the derivative function or simply the derivative, that gives the slope of the original function at any given point. In the context of our exercise, finding the derivative of the function \( f(x)=\frac{x}{\sqrt{2x-1}} \) is the first step in determining where the function has horizontal tangent lines, which occur where the derivative value is zero.

Quotient Rule

In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Specifically, if \( y = \frac{u}{v} \) where both \(u\text{ and } v \) are functions of \( x \) and are differentiable, then the derivative \( y' \) is given by the formula \( y' = \frac{v.u'-u.v'}{v^{2}} \).

This rule simplifies the process of differentiation when dealing with ratios and is particularly handy in our example where we have \( f(x) = \frac{x}{\sqrt{2x-1}} \). By identifying \( u \) as the numerator \( x \) and \( v \) as the denominator \( \sqrt{2x-1} \) and applying the quotient rule, we can find the derivative \( f'(x) \) necessary for locating horizontal tangents.

Finding Horizontal Tangents

A horizontal tangent line to a curve occurs at a point where the derivative of the function is zero. Visually, this is seen where the curve flattens out completely. In the process of finding horizontal tangents, the derivative of the function is set to zero because a horizontal line has a slope of zero. Solving for \( x \) when \( f'(x) = 0 \) provides us with the \( x \) coordinate(s) where the function has a horizontal tangent.

To complete the process, one must find the corresponding \( y \) coordinates by substituting the \( x \) values back into the original function \( f(x) \). The points with coordinates \( (x, f(x)) \) where the function has horizontal tangents are critically important for understanding the behavior of the function—they often identify maxima, minima, or inflection points on the graph.