Problem 23 - Calculus of a Single Variable

Question

In Exercises 5–24, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. $$ y=x^{5}-5 x $$

Step-by-Step Solution

Answer

The function \(y = x^{5}-5x\) has intercepts at x=0 and x = ±√{5}, a relative maximum at x = -1, a relative minimum at x = 1, and an inflection point at \(x = 0\). The function has no asymptotes.

Step 1: Find the intercepts

To find the intercepts, set \( y = 0 \) in the equation, which gives \( x^{5}-5x = 0 \). Factor out x to get \( x*(x^{4}-5) = 0 \). Solutions are x=0 and x = ±√{5}.

Step 2: Find the extrema

Take the first derivative of the function \( y' = 5x^{4}-5 = 5(x^{4}- 1) \). Set derivative equal to zero to get the critical points. So, \( x^{4} = 1 \), giving x = ±1 as critical points.

Step 3: Second derivative test

Calculate second derivative \( y'' = 20x^{3} \). Substitute critical points into second derivative. For x = -1, substituting in second derivative yields -20, so it’s a relative maximum. For x = 1, we get +20, so it’s a relative minimum.

Step 4: Finding inflection points

Setting second derivative equal to zero, we get \( 20x^{3}= 0 \), yielding a single inflection point at x = 0.

Step 5: Asymptotes

As x approaches infinity or -infinity, y also approaches infinity or -infinity. So, there are no asymptotes for the function.

Step 6: Verification using graphing utility

One can use graphing software to create the plot of the function. It should reflect the intercepts, relative extrema, inflection points and asymptotes found.

Key Concepts

Intercepts

Intercepts are points where the graph of a function crosses the axes. These points are crucial for sketching graphs as they give starting points for plotting.

For the function \( y = x^5 - 5x \), we found the intercepts by setting \( y = 0 \), giving us the equation \( x^5 - 5x = 0 \). By factoring, we get \( x(x^4 - 5) = 0 \).

This results in three intercepts: \( x = 0 \), \( x = \sqrt{5} \), and \( x = -\sqrt{5} \).

On a graph, you'll mark these points where the curve will pass through the x-axis.

Relative Extrema

Relative extrema include local maxima and minima in a function's graph. They depict the peaks and valleys, where the function changes direction.

For \( y = x^5 - 5x \), we found these by determining the critical points using the first derivative, \( y' = 5x^4 - 5 \).

Setting \( y' = 0 \) gave us critical points \( x = 1 \) and \( x = -1 \). Applying the second derivative test with \( y'' = 20x^3 \), at \( x = -1 \), we found a relative maximum since \( y''(-1) = -20 \), which is negative.

At \( x = 1 \), \( y''(1) = 20 \) is positive, showing a relative minimum.

These help identify where the graph rises or falls, crucial for understanding curve behavior.

Inflection Points

Inflection points are where the curvature of the graph changes. It's where the graph shifts from concave up to concave down or vice versa.

They are found using the second derivative, which for our function is \( y'' = 20x^3 \).

Setting \( y'' = 0 \) leads us to the inflection point \( x = 0 \). At this point, the graph changes its curvature.

Marking this point is essential as it informs us of the graph's change in concavity, enhancing the accuracy of the graph sketching.

Asymptotes

Asymptotes are lines that get infinitely close to the graph but never touch. They typically appear in rational functions.

Our function, \( y = x^5 - 5x \), lacks asymptotes because as \( x \) approaches infinity, \( y \) approaches infinity too.

This indicates that the graph extends indefinitely without leveling out near any line.

Understanding the absence of asymptotes helps confirm the function's end-behavior, crucial for completing your graph. Asymptotes often guide the overall shape but in this case, they do not influence the sketch.