Problem 68 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

The function \(f(x)\) is continuous in the interval \([a, b]\) and has values of the same sign on its end-points. Can one assert that there is no point in \([a, b]\) at which the function becomes zero?

Step-by-Step Solution

Answer

No, one cannot assert that there is a point in [a, b] at which the function becomes zero if the function is continuous in the interval and has values of the same sign on its end-points.

Step 1: Understand the Statement

The statement of the exercise talks about a function \(f(x)\) that is continuous on the interval \([a, b]\), with \(f(a)\) and \(f(b)\) having the same sign. The goal is to determine whether or not there can be a point in \([a, b]\) where the function has a value of zero.

Step 2: Apply the Intermediate Value Theorem (IVT)

The Intermediate Value Theorem states that if a function \(f\) is continuous on a closed interval \([a,b]\), and \(k\) is any value between \(f(a)\) and \(f(b)\), then there is at least one value \(c\) in the interval \([a, b]\) such that \(f(c) = k\).

Step 3: Apply the conditions of the theorem

In this case, since \(f(a)\) and \(f(b)\) are of the same sign, there is no value of \(k\) that can be zero, that falls between \(f(a)\) and \(f(b)\). That is, zero does not lie between two numbers of the same sign.

Step 4: Draw Conclusion

So, applying the Intermediate Value Theorem, it can be asserted that there is no point in \([a, b]\) at which the function \(f(x)\) becomes zero.

Key Concepts

Continuous Function

A continuous function is a function without any breaks or jumps within its domain. Imagine tracing a curve on a piece of paper without lifting your pencil. This is what it means for a function to be continuous over an interval, such as \([a, b]\). When a function is continuous on a closed interval like this, it has no abrupt changes, meaning its graph can be drawn in one unbroken stroke.

• Continuous functions are predictable in nature, which makes them easier to analyze.
• The Intermediate Value Theorem relies heavily on this predictability.

A function being continuous is a critical feature when using various mathematical theorems, including the Intermediate Value Theorem. It ensures that no values within the interval \([a, b]\) are skipped.

Zero of a Function

The zero of a function is an input value, say \(c\), where the output of the function is zero. In mathematical terms, if \(f(c) = 0\), then \(c\) is a zero or root of the function. Finding such zeros often tells us where a function crosses or touches the x-axis in a graph.

• If a function is continuous and has different signs at the end-point values, it suggests there's at least one zero between these two points.
• For example, if \(f(a) < 0\) and \(f(b) > 0\), then there must be a point \(c\) where \(f(c) = 0\).

In our specific problem, because \(f(a)\) and \(f(b)\) share the same sign, there is no way for the function to transition through zero between \([a, b]\). The graph does not touch or cut the x-axis anywhere on this interval.

Sign of a Function

The sign of a function on an interval indicates whether its outputs are positive, zero, or negative throughout that particular range. \(f(x)\)'s sign can tell us a lot about how the function behaves.

• When a function's sign is positive on an interval, its outputs are above the x-axis.
• If the sign is negative, then the function's outputs lie below the x-axis.

For the function \(f(x)\) across the interval \([a, b]\), if its values at both ends \(f(a)\) and \(f(b)\) are the same (both positive, or both negative), the function likely retains this sign across the entire interval. Thus, it becomes impossible for the function to be zero at any point within \([a, b]\), because the function doesn't switch from one side of the x-axis to the other.