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Chapter 4: Problem 44

Using the intermediate value theorem, determine, if possible, whether the function \(f\) has a real zero between a and \(b\). $$f(x)=2 x^{5}-7 x+1 ; a=1, b=2$$

Short Answer

Expert verified
Yes, there is a real zero between 1 and 2.

Step by step solution

01

Understand the Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on the interval \([a,b]\), and if \(f(a)\) and \(f(b)\) have opposite signs, then there is at least one real zero between \(a\) and \(b\).
02

Verify the continuity of the function

The given function is \(f(x)=2x^{5} - 7x + 1\), which is a polynomial. Polynomial functions are continuous for all real numbers. Therefore, \(f(x)\) is continuous on the interval \[1, 2\].
03

Calculate \(f(a)\)

Calculate \(f(1)\):\[ f(1) = 2(1)^{5} - 7(1) + 1 = 2 - 7 + 1 = -4 \] \Therefore, \(f(1) = -4\).
04

Calculate \(f(b)\)

Calculate \(f(2)\):\[ f(2) = 2(2)^{5} - 7(2) + 1 = 2(32) - 14 + 1 = 64 - 14 + 1 = 51 \] \Therefore, \(f(2) = 51\).
05

Analyze the sign change

We have \(f(1) = -4\) and \(f(2) = 51\). Since \(-4 < 0\) and \(51 > 0\), \(f(a)\) and \(f(b)\) have opposite signs.
06

Conclusion using the Intermediate Value Theorem

Since \(f(x)\) is continuous on \[1, 2\] and \(f(1)\) and \(f(2)\) have opposite signs, by the Intermediate Value Theorem, there is at least one real zero between \(a = 1\) and \(b = 2\).

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Key Concepts

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

Polynomial Functions
A polynomial function is an expression involving terms that are non-negative integer powers of a variable. For example, the function given in the exercise, \(f(x) = 2x^5 - 7x + 1\), is a polynomial. Polynomial functions can have various degrees based on the highest power of the variable. Here, the highest power is 5, so it is a fifth-degree polynomial.

Some properties of polynomial functions include:
  • They are smooth and continuous.
  • They can have multiple turning points and real zeros.
  • The behavior of the function at extreme values depends on the leading term.
Understanding polynomial functions is key to grasping higher-level concepts in calculus and algebra.
Continuity
A function is said to be continuous if there are no breaks, holes, or jumps in its graph. This means you can draw the graph without lifting your pen off the paper. In this exercise, the function given is a polynomial. Polynomial functions are always continuous for all real numbers.

The importance of continuity lies in its applications, such as:
  • Using the Intermediate Value Theorem to find zeros of a function.
  • Ensuring smooth transitions in graphs and data modeling.
  • Working with limits and derivatives in calculus.
Being aware of the continuity of a function helps in understanding if certain theorems, like the Intermediate Value Theorem, can be applied.
Real Zeros
A real zero of a function is a value of \(x\) that makes the function equal to zero, i.e., \(f(x) = 0\). Finding real zeros is important in solving equations and understanding the behavior of functions. In the context of the Intermediate Value Theorem, if a function changes signs over an interval, it guarantees at least one real zero in that interval.

According to the Intermediate Value Theorem, for the function \(f(x)\) given:
\[f(a) = -4 \quad \text{and} \quad f(b) = 51\]
Here, \(a = 1\) and \(b = 2\), and since \(f(a)\) and \(f(b)\) have opposite signs, there is at least one real zero between 1 and 2. This understanding is crucial for solving real-world problems that involve finding values within a specific range.

Thus, grasping the concept of real zeros helps in multiple areas such as:
  • Solving polynomial equations.
  • Determining intercepts of graphs.
  • Analyzing and predicting behaviors of functions.

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Most popular questions from this chapter

Fill in the blank with the correct term. Some of the given choices will not be used. Others will be used more than once. $$\begin{array}{ll}x \text { -intercept } & \text { midpoint formula } \\ y \text { -intercept } & \text { horizontal lines } \\ \text { odd function } & \text { vertical lines } \\ \text { even function } & \text { point-slope equation } \\ \text { domain } & \text { slope-intercept equation } \\ \text { range } & \text { difference quotient } \\ \text { slope } & f(x)=f(-x) \\\ \text { distance formula } & f(-x)=-f(x)\end{array}$$ The _______ is \(\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)\).

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What does Descartes' rule of signs tell you about the number of positive real zeros and the number of negative real zeros of the function? $$h(x)=6 x^{7}+2 x^{2}+5 x+4$$

Fill in the blank with the correct term. Some of the given choices will not be used. Others will be used more than once. $$\begin{array}{ll}x \text { -intercept } & \text { midpoint formula } \\ y \text { -intercept } & \text { horizontal lines } \\ \text { odd function } & \text { vertical lines } \\ \text { even function } & \text { point-slope equation } \\ \text { domain } & \text { slope-intercept equation } \\ \text { range } & \text { difference quotient } \\ \text { slope } & f(x)=f(-x) \\\ \text { distance formula } & f(-x)=-f(x)\end{array}$$ The _____ of the line with slope \(m\) passing through \(\left(x_{1}, y_{1}\right)\) is \(y-y_{1}=m\left(x-x_{1}\right)\).
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