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Chapter 3: Problem 49

Determine whether the intermediate value theorem guarantees that the function has a zero on the given interval. \(f(x)=2 x^{3}-7 x^{2}-14 x+30\) a. [1,2] b. [2,3] c. [3,4] d. [4,5]

Short Answer

Expert verified
The intermediate value theorem guarantees a zero in intervals [1, 2] and [4, 5].

Step by step solution

01

Evaluate the function at the given interval endpoints for [1, 2]

Calculate the values of the function at the endpoints of the interval [1, 2]: \[ f(1) = 2(1)^3 - 7(1)^2 - 14(1) + 30 \] \[ f(1) = 2 - 7 - 14 + 30 = 11 \]\[ f(2) = 2(2)^3 - 7(2)^2 - 14(2) + 30 \] \[ f(2) = 16 - 28 - 28 + 30 = -10 \] Check if one value is positive and the other is negative.
02

Apply the intermediate value theorem for [1, 2]

Because \( f(1) = 11 \) and \( f(2) = -10 \) have opposite signs, according to the intermediate value theorem, there is at least one value \( c \) in the interval [1, 2] where \( f(c) = 0 \). Hence, a zero is guaranteed in this interval.
03

Evaluate the function at the given interval endpoints for [2, 3]

Calculate the values of the function at the endpoints of the interval [2, 3]: \[ f(2) = -10 \] \( (Already calculated in Step 1) \)\[ f(3) = 2(3)^3 - 7(3)^2 - 14(3) + 30 \] \[ f(3) = 54 - 63 - 42 + 30 = -21 \] Check if the values have opposite signs.
04

Apply the intermediate value theorem for [2, 3]

Because \( f(2) = -10 \) and \( f(3) = -21 \) do not have opposite signs, the intermediate value theorem does not guarantee a zero in this interval.
05

Evaluate the function at the given interval endpoints for [3, 4]

Calculate the values of the function at the endpoints of the interval [3, 4]: \[ f(3) = -21 \] \( (Already calculated in Step 3) \) \[ f(4) = 2(4)^3 - 7(4)^2 - 14(4) + 30 \] \[ f(4) = 128 - 112 - 56 + 30 = -10 \] Check if the values have opposite signs.
06

Apply the intermediate value theorem for [3, 4]

Because \( f(3) = -21 \) and \( f(4) = -10 \) do not have opposite signs, the intermediate value theorem does not guarantee a zero in this interval.
07

Evaluate the function at the given interval endpoints for [4, 5]

Calculate the values of the function at the endpoints of the interval [4, 5]: \[ f(4) = -10 \] \( (Already calculated in Step 5) \) \[ f(5) = 2(5)^3 - 7(5)^2 - 14(5) + 30 \] \[ f(5) = 250 - 175 - 70 + 30 = 35 \]Check if the values have opposite signs.
08

Apply the intermediate value theorem for [4, 5]

Because \( f(4) = -10 \) and \( f(5) = 35 \) have opposite signs, the intermediate value theorem guarantees that there is at least one value \( c \) in the interval [4, 5] where \( f(c) = 0 \). Hence, a zero is guaranteed in this interval.

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Key Concepts

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

Polynomial Functions
Polynomial functions are expressions that involve variables raised to non-negative integer exponents. They are of the form: f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0, where each coefficient a_i is a real number. The degree of the polynomial is the highest exponent, and its leading coefficient is the coefficient of the highest power.
Polynomials can be classified by their degree:
  • Linear polynomial: Degree 1, e.g., f(x) = 3x + 2
  • Quadratic polynomial: Degree 2, e.g., f(x) = 2x^2 + 3x + 1
  • Cubic polynomial: Degree 3, e.g., f(x) = x^3 - 4x + 6
When working on polynomial functions, you often need to find their roots or zeros, values of x where f(x) = 0.
Interval Evaluation
Interval evaluation involves calculating the value of a function at the endpoints of a given interval. It's commonly used with the Intermediate Value Theorem (IVT). Here鈥檚 how you do it:
  • Identify the interval, for example, [a, b]
  • Calculate f(a) and f(b)
  • Use the results to check if f(a) and f(b) have opposite signs
For instance, in the exercise above, evaluating f(x) = 2x^3 - 7x^2 - 14x + 30 at f(1) = 11 and f(2) = -10 shows that the function changes signs between x = 1 and x = 2. This suggests there is at least one zero in the interval.
Finding Zeros of Functions
Finding the zeros of a function involves identifying the values of x where f(x) = 0. There are several methods for finding zeros:
  • Factoring: Splitting the polynomial into products of simpler polynomials.
  • Graphical Method: Plotting the function and finding the x-coordinates where the graph intersects the x-axis.
  • Numerical Methods: Using algorithms like Newton-Raphson for more complicated polynomials.
Using the Intermediate Value Theorem (IVT) is another method for proving the existence of zeros over an interval. In the given problem, IVT is used on different intervals to determine if there is a zero by checking if the function values at the ends of the interval differ in sign.
College Algebra
College Algebra is a foundational mathematics course that covers a variety of essential topics. It typically includes:
  • Polynomial and Rational Functions
  • Exponential and Logarithmic Functions
  • Systems of Equations and Inequalities
  • Sequences and Series
  • Basic Matrix Operations
Understanding these topics is crucial for success in higher-level mathematics and many other fields. The Intermediate Value Theorem (IVT) and techniques for interval evaluation are classic examples of College Algebra applications. They teach how to determine the existence of solutions using fundamental properties of continuous functions and the behavior of polynomials over specific intervals.

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Most popular questions from this chapter

Write an equation of a function that meets the given conditions. Answers may vary. \(x\) -intercept: \(\left(\frac{4}{3}, 0\right)\) vertical asymptotes: \(x=-3\) and \(x=-4\) horizontal asymptote: \(y=0\) \(y\) -intercept: (0,-1)

The procedure to solve a polynomial or rational inequality may be applied to all inequalities of the form \(f(x)>0, f(x)<0,\) \(f(x) \geq 0,\) and \(f(x) \leq 0 .\) That is, find the real solutions to the related equation and determine restricted values of \(x .\) Then determine the sign of \(f(x)\) on each interval defined by the boundary points. Use this process to solve the inequalities. $$ \frac{1}{\sqrt{x-3}-5} \leq 0 $$

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Let \(n\) be a positive even integer. Determine the greatest number of possible imaginary zeros of \(f(x)=x^{n}-1\).
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