Question

Find the \(x\) -values where the graph of the function has a horizontal tangent line. \(f(x)=x \sin x\) on [-1,1].

Short answer

Find where the derivative is zero on [-1,1].

Step-by-step solution

Understand the Problem

To find where the graph of a function has a horizontal tangent line, we need to determine where its derivative is equal to zero. A horizontal tangent implies a slope of zero.

Find the Derivative

First, we calculate the derivative of the given function. The function is given as: \[ f(x) = x \sin x \]To find the derivative \(f'(x)\), we need to use the product rule, which states that \((uv)' = u'v + uv'\). Here, let \(u = x\) and \(v = \sin x\).

Key concepts

Derivative

A derivative measures how a function changes as its input changes. It essentially tells us the slope of the tangent line to the function at any given point. When we seek to identify horizontal tangent lines, we are looking for points where this slope is zero. This means we need to find where the derivative of the function equals zero.

For the function given in the exercise, finding the derivative is the key step to identifying the points of interest. If the derivative of a function, noted as \(f'(x)\), equals zero at a certain \(x\)-value, the tangent line at that point is horizontal. Finding these specific \(x\)-values is what the exercise asks us to do.

Product Rule

When functions are multiplied together, like in the case of \(f(x) = x \sin x\), we use the product rule to find the derivative. The product rule is a formula that helps us differentiate the product of two functions. It can be expressed as:

• If \(u(x)\) and \(v(x)\) are two functions, the derivative of their product \(uv\) is \((uv)' = u'v + uv'\).

In this exercise, you are given \(u = x\) and \(v = \sin x\). Understanding the product rule allows us to find \(f'(x)\). First, differentiate \(u\) and \(v\):

• \(u' = 1\)
• \(v' = \cos x\)

Now, apply the product rule:

• \(f'(x) = 1 \cdot \sin x + x \cdot \cos x = \sin x + x \cos x\)

With \(f'(x)\) calculated, we can now find the \(x\) values where the graph of the function has a horizontal tangent by setting \(f'(x) = 0\). This means solving the equation \(\sin x + x \cos x = 0\).

Graph of a Function

The graph of a function represents the relationship between the input (often \(x\)) and output (often \(f(x)\)). It provides a visual representation to better understand the behavior of a function, such as where tangent lines are horizontal, indicating points of slope zero.

For \(f(x) = x \sin x\), we are looking at how this sinusoidal function behaves over the interval \([-1, 1]\). The function’s graph can have regions where it is increasing or decreasing, and points where the rate of change (or slope) is zero—these are the horizontal tangents.

When analyzing a graph, finding where horizontal tangents occur can help understand key features like peaks and troughs. In the interval \([-1, 1]\), setting \(f'(x) = 0\) specifies these points. By locating these \(x\)-values visually or algebraically, one gains insight into the structure and important characteristics of the function's graph within the specified domain.