Problem 46 - Calculus

Question

Are the statements true or false? Give an explanation for your answer. If \(f\) is continuous on the interval [0,10] and \(f(0)=0\) and \(f(10)=100,\) then \(f(c)\) cannot be negative for \(c\) in [0,10]

Step-by-Step Solution

Answer

The statement is false.

Step 1: Understanding Continuity

We are given that the function \(f\) is continuous on the interval \([0, 10]\). This means there are no breaks, jumps, or holes in the graph of the function within this interval.

Step 2: Analyzing Given Points

The function intercepts are \(f(0) = 0\) and \(f(10) = 100\). These points suggest that at \(x = 0\), the value of the function is 0, and at \(x = 10\), the value of the function is 100.

Step 3: Applying the Intermediate Value Theorem

By the Intermediate Value Theorem (IVT), if \(f\) is continuous on \([0, 10]\) and \(f(0) = 0\) and \(f(10) = 100\), then \(f\) must take every value between 0 and 100 at least once within the interval \([0, 10]\). However, IVT does not guarantee that the function must stay positive at all points between these values.

Step 4: Considering Negative Values

While the Intermediate Value Theorem ensures \(f\) takes intermediate values, it does not prevent \(f\) from dipping below zero, becoming negative for some \(c\) in \([0, 10]\), as long as it returns to 100 at \(x=10\). Thus, \(f(c)\) could be negative at some point in \([0, 10]\), as continuity only means no sudden jumps but not restrictions on going negative within the given interval.

Key Concepts

Continuity

Continuity in mathematics is a fundamental concept that dictates how a function behaves over an interval. A function is said to be continuous on a given interval if there are no breaks, jumps, or holes in its graph within that interval. This concept is like drawing a line on a paper without lifting the pencil. For example, if a function is continuous on the interval \([0, 10]\), it means that the function smoothly connects from the point \(x = 0\) to \(x = 10\), without any interruptions.
Moreover, the notion of continuity implies that small changes in the input result in small changes in the output, portraying a stable and predictable function behavior. The absence of discontinuities is crucial to apply certain mathematical theorems, like the Intermediate Value Theorem, which further analyze function behavior.

Negative Values

Negative values in a function refer to points where the function outputs less than zero. When analyzing functions, it is important to note that being continuous does not inherently prevent a function from taking negative values.

• Functions can move both above and below the x-axis while maintaining continuity.
• Even if a continuous function starts at zero and ends at a high positive value, there could still be parts of the graph that dip below zero unless other conditions are specified.

These dips below the x-axis represent places where the function has negative values. Understanding this helps clarify that the Intermediate Value Theorem focuses on reaching all intervening values but does not restrict excursions into negative values if the function moves to non-negative values subsequently.

Function Behavior

The behavior of a function tells us how it moves and changes across its domain. For continuous functions, this means:

• Eliminating unexpected jumps and breaks.
• Smoothing transitions between values.

In our exercise, the function starts at zero when \(x = 0\) and reaches 100 at \(x = 10\). While continuity ensures these values are hit without disruption, it doesn't dictate the path in between. Therefore, even though the function must smoothly progress from 0 to 100, it might dip into negative territory if there are no other specific constraints.
Understanding function behavior—how steeply it rises or falls, any peaks or troughs it might have—sheds light on potential places where the function might assume negative values before completing its ascent or descent across the interval.

Value Theorem Implications

The Intermediate Value Theorem (IVT) has key implications for continuous functions. The theorem states that for any value between the function's start and end points, there must be some input where the function takes on that value. Applying this to a function over an interval guarantees that not only is every intermediate value achieved, but it also highlights the underdefined aspects concerning negativity.

• A continuous function must hit every point between \(f(0) = 0\) and \(f(10) = 100\).
• However, IVT does not avoid values less than zero if the function passes through zero and 100.

Thus, while the theorem requires the function to reach all values between 0 and 100, it does not preclude negative values. There could indeed be points where the function dips below zero as long as it covers every value between the specified ends—a critical takeaway when interpreting the outcome of the Intermediate Value Theorem.