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Chapter 0: Problem 14

Sketch a graph of the function showing all extreme, intercepts and asymptotes. $$f(x)=12-x^{5}$$

Short Answer

Expert verified
The function has one extreme at \( x=0 \), x-intercepts at \( x=\sqrt[5]{12} \) and \( x=-\sqrt[5]{12} \), and a y-intercept at (0, 12). It has no asymptotes. The graph for this function will show all these points and behaviours.

Step by step solution

01

Identify Extremes

First, find the function's derivative, \( f'(x) \), because extremes occur where the derivative is zero or undefined. So, \( f'(x) = -5x^{4} \).Setting \( f'(x) = 0 \) and solving for \( x \) gives \( x=0 \) as the only extreme point.
02

Calculate Intercepts

Second, to find the x-intercepts, set the function equal to zero and solve for \( x \). This gives \( 12-x^{5} = 0 \). Thus \( x=\sqrt[5]{12} \) and \( x = -\sqrt[5]{12} \) are the x-intercepts. The y-intercept is found by setting \( x=0 \) in the function, which results in \( f(0)=12 \). Thus, the y-intercept is at (0, 12).
03

Find Asymptotes

There is no value of \( x \) that makes the denominator zero, and the function doesn't approach a certain value as \( x \) reaches infinity; hence, there are no asymptotes.
04

Sketch the Graph

Combine all the above findings to sketch the graph. Ensure to mark the point of the extreme, the intercepts, and note that there are no asymptotes.

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Key Concepts

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

Extreme Points
Extreme points on a graph are where the curve either reaches a peak (maximum) or a valley (minimum). To find these points for a given function, we use derivatives.
  • First, compute the derivative of the function. For example, if you have a function like \( f(x) = 12 - x^5 \), calculate \( f'(x) \).
  • In this example, the derivative is \( f'(x) = -5x^4 \).
  • Set the derivative equal to zero to find critical points. Solving \( -5x^4 = 0 \) gives \( x = 0 \).
  • This means the extreme point is at \( x = 0 \).
It's important to note whether this extreme point is a maximum or minimum.
  • Since the derivative is zero, you can test surrounding values of \( x \) to determine the nature of the turn at \( x = 0 \).
  • In this case, around \( x = 0 \), the function decreases and increases symmetrically, indicating a local plateau.
Intercepts
Intercepts are points where the graph intersects the axes. They help describe where a curve passes through or reaches the x-axis or y-axis. Let's look at the intercepts of our function:
  • **X-intercepts**, found by setting the function equal to zero and solving for \( x \). For \( f(x) = 12 - x^5 \), solve \( 12 - x^5 = 0 \). This results in the solutions \( x = \sqrt[5]{12} \) and \( x = -\sqrt[5]{12} \).
  • The **Y-intercept** occurs where \( x = 0 \). For our function, substitute 0 for \( x \) in \( f(x) = 12 - x^5 \). This results in \( f(0) = 12 \), giving a y-intercept at \( (0, 12) \).
Understanding intercepts helps you anchor your sketch since these are confirmed points through which the graph must pass.
Asymptotes
Asymptotes are lines that a graph approaches but never actually reaches. They show long-term behavior as \( x \) becomes very large (positive or negative). Asymptotes can be horizontal, vertical, or oblique (slanted). For the function \( f(x) = 12 - x^5 \), we determine if asymptotes exist:
  • No **vertical asymptotes** are present because there鈥檚 no denominator that would make the function undefined.
  • No **horizontal asymptotes** as the degree of \( x \) is higher in the polynomial than any constant term. The function鈥檚 value will increase or decrease without bound, as \( x^5 \) dominates the equation as \( x \) grows larger.
In some cases, functions will approach certain lines as \( x \) becomes very large, but for \( f(x) = 12 - x^5 \), the dominance of \( x^5 \) implies no asymptotic behavior.
Derivative
Derivatives are fundamental to finding the behavior of functions, including extreme points and slope calculations. A derivative is a function showing the rate of change of another function.**Finding Derivatives:**
  • Take the derivative of the function to understand how the output value changes with respect to \( x \). This often involves applying power rules, such as \( \, \frac{d}{dx} x^n = nx^{n-1} \, \).
  • For example, the derivative of \( f(x) = 12 - x^5 \) is \( f'(x) = -5x^4 \). This results from applying the power rule to \( -x^5 \).
**Using Derivatives:**
  • The derivative \( f'(x) \) indicates where the slope of the tangent is zero (extreme points), or where it becomes undefined (sharp corners, cusps).
  • In our function, since the derivative \( -5x^4 \) never becomes undefined, no sharp corners or cusps occur. The point where it equals zero, \( x = 0 \), identifies an extreme point.
Derivatives not only help determine extreme points, they are also crucial for understanding increasing and decreasing behaviors as well as curvature of the graph.

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