Question

Test your understanding of the Intermediate Value Theorem. Let \(g\) be continuous on [-3,7] where \(g(0)=0\) and \(g(2)=\) 25. Does a value \(-3

Short answer

Yes, such a value \( c \) exists by the Intermediate Value Theorem.

Step-by-step solution

Understanding the Intermediate Value Theorem

The Intermediate Value Theorem (IVT) states that if a function \( g \) is continuous on a closed interval \([a, b]\), and \( N \) is any number between \( g(a) \) and \( g(b) \), then there exists at least one number \( c \) in the interval \((a, b)\) such that \( g(c) = N \). Our task is to use this theorem to determine if there exists a number \( c \) such that \( g(c) = 15 \).

Obtain Interval Information

First, identify the values \(g(0)\) and \(g(2)\). We know from the problem that \(g(0) = 0\) and \(g(2) = 25\). Therefore, we will consider the interval \([0, 2]\) where the function is continuous.

Applying IVT to the Interval [0, 2]

According to IVT, since \(g\) is continuous on \([0, 2]\) and \(15\) is a number between \(g(0) = 0\) and \(g(2) = 25\), there must exist some \( c \) in \((0, 2)\) such that \( g(c) = 15 \).

Conclusion on Interval [-3, 7]

Since the theorem applies on a smaller interval \( [0, 2] \) where the condition is satisfied for \( g(c) = 15 \), it also holds on the larger interval \( [-3, 7] \), because \( [-3, 7] \) includes \([0, 2]\). Hence, \( c \) exists within \(-3 < c < 7 \) such that \( g(c) = 15 \).

Key concepts

Understanding Continuous Functions

A continuous function is a type of function where small changes in input result in small changes in output. This smooth transition means that the graph of the function does not have any breaks, jumps, or holes within its domain. In the context of the Intermediate Value Theorem (IVT), this concept is crucial, as the theorem depends on the function being continuous over a certain interval to ensure the existence of certain function values. For a function like \( g \) in the given problem, being continuous implies no sudden leaps from one y-value to another as x-values move through the interval

• It allows us to apply the Intermediate Value Theorem confidently, knowing that within any sub-interval, the function will pass through every value between its endpoints.
• Since the function \( g \) is continuous between \(-3\) and \(7\), any number between the function values at these points must occur somewhere in this interval.

Defining a Closed Interval

A closed interval, denoted as \([a, b]\), includes all the points between \(a\) and \(b\), including the endpoints themselves. This is different from open intervals, which do not include the endpoints, and is usually written as \((a, b)\). In analysis, working within a closed interval like \([-3, 7]\) ensures that we are considering every point, including the start and end, emphasizing that there are no gaps.

• It is important for checking all values the function might take across the interval.
• Ensures the Intermediate Value Theorem can be applied as it requires continuity across such intervals.

This becomes particularly relevant to ensure the function \(g\) reaches the desired number between known values, across its domain.

Exploring the Role of Function Values

Function values are the output values a function generates for specific input values. They are crucial for applying the Intermediate Value Theorem, which states that for a continuous function on a closed interval, every value between the function’s endpoints must also be achieved by the function.

• In our exercise, knowing that \(g(0) = 0\) and \(g(2) = 25\) provides the boundary function values needed to determine if \(g(c) = 15\) could exist.
• Since \(15\) lies between the known function values of \(0\) and \(25\), the theorem guarantees that \(15\) occurs somewhere within this interval as a result of the pipeline of values the function passes through.

Every step in reasoning involves considering how function values interact within a given range, as ensuring continuity provides a sort of pathway or bridge across these values.

The Existence of Solutions Through IVT

One of the profound implications of the Intermediate Value Theorem is the assurance of the existence of solutions for certain equations, specifically those that can be represented by continuous functions over a closed interval. The IVT assures us that if the function is continuous and the desired outcome lies between two known outputs, then a solution exists where the function takes that value.

• In the problem, because \(g\) is continuous from \(0\) to \(2\), and because \(15\) is between \(0\) and \(25\), we know there must be a \(c\) within \((0, 2)\) such that \(g(c) = 15\).
• This logic extends our consideration throughout the entire \([-3, 7]\) interval, showing there exists at least one \(c\) where \(-3 < c < 7\) yielding \(g(c) = 15\).

This theorem effectively bridges the known and unknown, providing a powerful analytical tool to ascertain the existence of solutions within specified domains.