Problem 36 - 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. $$\sqrt{3} \cos x+\sin x$$

Step-by-Step Solution

Answer

Critical points: \( n\pi + \frac{\pi}{6} \). Inflection points: \( n\pi - \frac{\pi}{3} \). Graph using these points.

Step 1: Identify the Function Components

The given function is \( f(x) = \sqrt{3} \cos x + \sin x \). Here, we can see that it is a combination of sine and cosine functions, which means it is a type of trigonometric function.

Step 2: Determine the Derivative of the Function

Find the first derivative of the function \( f(x) = \sqrt{3} \cos x + \sin x \). Differentiating term by term, we get:\[ f'(x) = -\sqrt{3} \sin x + \cos x \]. This derivative will help identify the critical points for relative extrema.

Step 3: Solve for Critical Points

Set \( f'(x) = 0 \) to find critical points:\[ -\sqrt{3} \sin x + \cos x = 0 \]Solve for \( x \):\[ \cos x = \sqrt{3} \sin x \]Divide both sides by \( \cos x \):\[ 1 = \sqrt{3} \tan x \]Therefore, \( \tan x = \frac{1}{\sqrt{3}} \), which happens at \( x = n\pi + \frac{\pi}{6} \) where \( n \) is an integer.

Step 4: Determine Relative Extrema

Plug the critical points back into the second derivative, or check sign changes in the first derivative across these points, to determine if they are maxima or minima. Here significant points within one full cycle \([0, 2\pi]\) include \( \frac{\pi}{6} \), \( \frac{7\pi}{6} \).

Step 5: Find the Second Derivative

Take the second derivative to identify inflection points:\[ f''(x) = -\sqrt{3} \cos x - \sin x \].

Step 6: Solve for Inflection Points

Set the second derivative to zero to find inflection points:\[ -\sqrt{3} \cos x - \sin x = 0 \]\[ \sqrt{3} \cos x = -\sin x \]Divide by \( \cos x \):\[ \sqrt{3} = -\tan x \]So, \( \tan x = -\sqrt{3} \), which gives \( x = n\pi - \frac{\pi}{3} \) where \( n \) is an integer.

Step 7: Construct the Graph

Consider the solutions to both the first and second derivatives for graphing and label the points found: \( \frac{\pi}{6} \), \( \frac{7\pi}{6} \), etc., as well as the inflection points. Input these into a graphing utility to visualize the waveform, ensuring to plot it over at least one complete cycle \([0, 2\pi]\).

Key Concepts

Derivatives

To understand how a function changes, we study its derivatives. Think of derivatives as the mathematical equivalent of taking the pulse of a function. They help us know how fast something is changing at any given point.
For our function, the first derivative is given by:

• \[ f'(x) = -\sqrt{3} \sin x + \cos x \]
  This derivative tells us how the steepness of our trigonometric function evolves as x changes.

By setting the derivative to zero, we find the critical points, the spots where our function might turn around—like a roller coaster reaching its peak or lowest point. Remember, finding the derivative is not just limited to mere computation; it reveals much about the behavior of a function!

Relative Extrema

Relative extrema are the high and low points in a particular section of a graph. They are where the function reaches a local maximum or minimum. If you think about hiking in hilly terrain, these points are akin to the hilltops and valley bottoms you encounter.
To find these in our function, we should first find the critical points by solving:

• \[ -\sqrt{3} \sin x + \cos x = 0 \]

After finding the critical points, these values are substituted back into the original function or checked in the derivative's vicinity for sign changes. Such investigation will tell us whether these points correspond to maxima or minima over the cycle \([0, 2\pi]\). This approach ensures a thorough understanding of the function's behavior.

Inflection Points

Inflection points are locations on the graph where the curve changes its direction of bending. It's like the crestline changing from a concave-up shape, like a bowl, to concave-down, like an arch. We identify these points by analyzing the function's second derivative:

• \[ f''(x) = -\sqrt{3} \cos x - \sin x \]

By setting this second derivative to zero, we effectively pinpoint the spots where the graph's curvature shifts. Solve \( -\sqrt{3} \cos x - \sin x = 0 \) to find those key values. Such computations help detect if the curve's bending transitions, providing depth to the graph's analysis.

Graphing Utility

A graphing utility is your virtual sketchpad for mathematics. It's a tool that allows us to visualize functions accurately and efficiently. With our trigonometric function, using a graphing utility means we can confidently interpret the location of all critical points, relative extrema, and inflection points.
Inputting our function into this tool will reveal the complete cycle from \(0\) to \(2\pi\), showing a vivid picture of hilly maxima and minima. This powerful visual aid backs up our mathematical findings and offers an immediate, clear view of where the function changes direction. Furthermore, you can check the consistency of your calculated critical and inflection points by examining how these match up on the plotted curve.

Critical Points

Critical points are crucial in determining where a function might experience important moments of change. They are identified where the derivative equals zero or doesn't exist. In our example, they occur where:

• \[ -\sqrt{3} \sin x + \cos x = 0 \]

Solving this means finding where the slope is zero, indicating maximum or minimum points in the context of our trigonometric journey. These critical points are \(x = n\pi + \frac{\pi}{6}\). They allow us to craft a roadmap on the function's behavior over a complete period. Verifying how these points act within their neighborhoods helps solidify our understanding of the function's narrative over its ongoing journey. Engage deeply with the function's rhythm, and you'll appreciate the pattern it portrays.