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Chapter 3: Problem 21

Determine all significant features by hand and sketch a graph. $$f(x)=x^{3 / 4}-4 x^{1 / 4}$$

Short Answer

Expert verified
The function \(f(x) = x^{3 / 4}-4 x^{1 / 4}\) has roots at x = 0 and x = 16, a y-intercept at (0,0), a vertical asymptote at x = 0, and a minimum point at x = 1. The function's graph increases from negative infinity to 0 at the y-intercept, decreases until reaching a minimum point at x = 1, then increases again, crossing the x-axis at x = 16.

Step by step solution

01

Identify the Roots

To find the roots of the function, set the function equal to zero and solve for x: \(x^{3/4} - 4x^{1/4} = 0\). This yields the roots of x = 0 and x = 16.
02

Determine the Y-intercept

The y-intercept is found by setting x=0 in the function, this yields \(y = f(0)\), thus the y-intercept is at the point (0,0).
03

Identify the Asymptote

Since the function is defined for all real x > 0, and the function approaches negative infinity as x approaches 0, we have a vertical asymptote at x = 0.
04

Identify Extrema Points

To find extrema points, first find the derivative of the function \(f'(x) = (3/4)x^{-1/4} - (x^{-3/4})\), then set this to zero to find critical points. Upon solving, only one critical point is found at x = 1, which gives a minimum point.
05

Sketch the Graph

Combine all the obtained information to sketch the graph. The curve increases from negative infinity to 0 at the y-intercept, decreases until the minimum point at x = 1, and then increases again from there. It crosses the x-axis again at x = 16.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Roots of Functions
Understanding the roots of a function is crucial for graph sketching. The roots, or zeroes, of a function are the x-values where the function equals zero. To find these, set the function equal to zero and solve for x. For the function \(f(x) = x^{3/4} - 4x^{1/4}\), we equate it to zero:
  • Firstly, isolate the terms: \(x^{3/4} = 4x^{1/4}\).
  • Simplify by dividing both sides by \(x^{1/4}\): \(x^{1/2} = 4\).
  • To get x, square both sides: \(x = 16\).
This tells us that the roots are at \(x = 0\) and \(x = 16\). Roots are important because they tell us where the graph crosses the x-axis. Identifying roots helps in plotting key points and understanding the function's behavior across the x-axis.
Y-intercept
The y-intercept is the point where the graph intersects the y-axis. To find it, substitute \(x = 0\) into the function, because at this point, the x-coordinate is zero. For our function, when \(x = 0\), we have:
  • Substitute into the function: \(f(0) = 0^{3/4} - 4(0)^{1/4}\)
  • Both terms are zero, giving \(f(0) = 0\).
Thus, the y-intercept is at the origin, \((0, 0)\). This is significant because it gives a starting point for sketching the graph. Knowing this intercept helps us, especially when we're quickly plotting the function's behavior in the vertical direction.
Vertical Asymptotes
Vertical asymptotes are lines that a graph approaches but never touches or crosses. They're seen where the function grows infinitely large or small as \(x\) approaches a particular value. In rational functions, these occur where the denominator is zero, but in our case, we look at the behavior near \(x = 0\). Here:
  • The function \(f(x) = x^{3/4} - 4x^{1/4}\) is defined for \(x > 0\).
  • As \(x\) approaches 0 from the positive side, the function heads towards negative infinity.
Hence, we recognize a vertical asymptote at \(x = 0\). Visualizing this on the graph, it provides a guide for how the curve behaves as \(x\) gets very small but positive, helping us trace the graph correctly.
Extrema Points
Extrema points are the peaks and troughs in a graph, representing maximum or minimum points. To identify them, calculate the derivative of the function and then find where this derivative is zero. Setting \(f'(x) = (3/4)x^{-1/4} - (x^{-3/4}) = 0\), we solve:
  • Combine like terms: \((3/4)x^{-1/4} = x^{-3/4}\).
  • Multiply through by \(x^{3/4}\): \( (3/4)x^{1/2} = 1 \).
  • Solve for \(x\) by squaring both sides: \(x = 1\).
The critical point at \(x = 1\) is verified to be a minimum by considering the sign change of the derivative. Knowing extrema helps shape the graph's appearance, indicating where it turns or flattens.

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