Problem 33 - Calculus A New Horizon

Question

Give a complete graph of the function, and identify the location of all relative extrema and inflection points. Check your work with a graphing utility. $$x+\sin x$$

Step-by-Step Solution

Answer

Relative extrema at \( x = (2n+1)\pi \); inflection points at \( x = n\pi \). Use graphing utility for confirmation.

Step 1: Find the derivative

To find the relative extrema and inflection points, we first need the derivative of the function. The given function is \( f(x) = x + \sin x \). The derivative is found using differentiation rules: \( f'(x) = 1 + \cos x \).

Step 2: Determine critical points

Set \( f'(x) = 0 \) to find the critical points. \( 1 + \cos x = 0 \) leads to \( \cos x = -1 \). The solutions to this equation are \( x = (2n+1)\pi \), where \( n \) is an integer.

Step 3: Second derivative for inflection points

Find the second derivative to locate inflection points: \( f''(x) = -\sin x \). Set \( f''(x) = 0 \) to find inflection points, leading to \( -\sin x = 0 \), so \( \sin x = 0 \). Solutions are \( x = n\pi \), where \( n \) is an integer.

Step 4: Classify critical points using the second derivative

Check the sign of \( f''(x) \) at critical points \( x = (2n+1)\pi \). If \( f''(x) > 0 \), the function has a relative minimum; if \( f''(x) < 0 \), a relative maximum. At these points, \( f''((2n+1)\pi) = -\sin((2n+1)\pi) = 0 \), indicating possible inflection points.

Step 5: Interpret inflection points, minimum, and maximum

Since \( \sin x = 0 \) at \( x = n\pi \), the concavity changes, confirming inflection points. Since \( f''((2n+1)\pi) = 0 \), confirm using plotting: relative extrema only at inflection points where the derivative indicates changes in slope.

Step 6: Graph the function using a graphing utility

Plot \( f(x) = x + \sin x \) to visualize the behavior of the function. Confirm the presence of critical points at \( x = (2n+1)\pi \) and inflection points at \( x = n\pi \). Note the linear component changes the expected behavior of a typical sine function.

Key Concepts

Critical Points

Critical points are crucial in understanding where functions may have local maximums or minimums. These points occur where the derivative of a function is equal to zero or is undefined.
In our case, the function is given by \( f(x) = x + \sin x \). To find its critical points, we take the derivative, which results in \( f'(x) = 1 + \cos x \).

By setting \( f'(x) = 0 \), we solve for \( \cos x = -1 \). This condition is met when \( x = (2n+1)\pi \), where \( n \) is an integer. Thus, these solutions represent the positions along the x-axis where the slope of \( f(x) \) is zero, indicating potential peaks or valleys.
Remember that identifying the critical points is the first step; verifying them involves further tests, such as examining concavity.

Derivative

A derivative is a mathematical tool used to find the rate of change of a function. It helps in understanding how the function's output changes with respect to changes in the input.
For the function \( f(x) = x + \sin x \), the derivative is calculated as \( f'(x) = 1 + \cos x \). This expression gives us valuable insights into the function's behavior.

When the derivative is zero, it highlights critical points. The form \( 1 + \cos x \) tells us how steep the slope is or how quickly the function's value changes.
Derivatives find their application in analyzing and graphing functions, especially when looking for extrema, concavity, and understanding the general shape of the graph.

Inflection Points

Inflection points are where the graph of a function changes its concavity from concave up to concave down or vice versa. To find these points, we examine where the second derivative is zero or undefined.
In this exercise, the second derivative of \( f(x) = x + \sin x \) is \( f''(x) = -\sin x \). Set \( f''(x) = 0 \) to locate inflection points, leading to \( \sin x = 0 \).

The solutions to this equation are \( x = n\pi \), indicating inflection points at multiples of \( \pi \), where \( n \) is an integer.

• These points mark where the function's curvature changes.
• Such changes can affect the smoothness and shape of a curve.

Depending on whether \( f''(x) \) is positive or negative, one can tell if the curve opens upwards or downwards.

Graphing Utility

Graphing utilities are powerful tools that allow us to visualize mathematical concepts, like functions and their critical behaviors, instantly.
When tasked with graphing \( f(x) = x + \sin x \), a graphing utility can quickly show the function's shape, critical points, and inflection points.

This function might exhibit unexpected behavior due to its linear component added to the sinusoidal wave. By using a graphing utility:

• Identify the periodic nature overlaying a linear slope.
• Verify critical points at \( x = (2n+1)\pi \).
• Note inflection points at \( x = n\pi \).

These tools give more than just visual confirmation; they help cross-verify analytical findings and deriving deep insights into the function’s dynamics.