Question

Assume \(\int\) is continuous on \([a, b]\). Assume also that \(\int_{a}^{b} f(x) g(x) d x=0\) for every function \(g\) that is continuous on \([\mathrm{a}, \mathrm{b}]\). Prove that \(\mathrm{f}(\mathrm{x})=0\) for all xin [a. b]

Short answer

Question: Prove that if the integral of a continuous function f(x) times the continuous function g(x) is zero over the interval [a, b], then the function f(x) must be zero over the whole interval. Short Answer: By using a proof by contradiction and defining a suitable continuous function g(x), we show that if our assumption of f(x) not being zero at any point in the interval [a, b] holds, it leads to a contradiction. Hence, we must reject our assumption and conclude that f(x) must be zero for all x in [a, b].

Step-by-step solution

Suppose that f(x) is not zero on the entire interval [a, b]

First, let's assume the opposite of the statement we want to prove, i.e., that f(x) is not zero for all x in [a, b]. Therefore, there must be at least one point 'c' in [a, b] where f(c) is not equal to zero.

Construct a suitable continuous function g(x)

Now, let's construct a continuous function g(x) which satisfies the conditions of the exercise. Define g(x) as follows: - g(x) = 1 if x = c, - g(x) is continuous and differentiable on [a, b], and g(x) = 0 at the endpoints a and b.

Integrate f(x)g(x) over [a, b]

Now, we want to calculate the integral of the product of f(x) and g(x). We know that the resulting function f(x)g(x) is also continuous on [a, b]. According to the problem statement, for any continuous function g(x), we have \(\int_{a}^{b} f(x) g(x) d x=0\). Therefore, for the g(x) that we have, we should also have: $$\int_{a}^{b} f(x) g(x) d x=0$$

Show the contradiction

Since f(c) ≠ 0 and g(c) = 1, the product f(c)g(c) is not equal to zero. This means at least at point 'c', the product of the two functions is non-zero. As g(x) is chosen in such a way that is continuous, differentiable, and has g(x) = 0 at the endpoints 'a' and 'b', the integral of the function f(x)g(x) should not be zero: $$\int_{a}^{b} f(x) g(x) d x ≠ 0$$ This result contradicts the problem statement.

Conclude the proof

Since our assumption in Step 1 has led to a contradiction, we must reject the assumption that f(x) is not zero everywhere on the interval [a, b]. This means that f(x) must be zero for all x in [a, b]. Therefore, we have proved that if \(\int_{a}^{b} f(x) g(x) d x=0\) for every continuous function g(x) on [a, b], then f(x) must be zero on the entire interval [a, b].

Key concepts

Integral Properties

Integral calculus is a cornerstone of mathematics, providing powerful tools for quantifying the accumulation of quantities like area, volume, and total change. Integral properties are formulas and rules that describe how integrals behave under certain mathematical operations and with respect to various kinds of functions. One of the most fundamental properties is the linearity of integrals, which states that the integral of a sum is the sum of the integrals, and the integral of a constant times a function is the constant times the integral of the function. This property can be formally expressed as:
\[ \int (af(x) + bg(x))dx = a\int f(x)dx + b\int g(x)dx \]
Moreover, there's the property of additivity over intervals, suggesting that the integral over a larger interval can be split into integrals over its subintervals.

• Additivity over intervals: If \( c \) lies between \( a \) and \( b \), then \[ \int_{a}^{b} f(x)dx = \int_{a}^{c} f(x)dx + \int_{c}^{b} f(x)dx. \]

Continuous Functions

A continuous function is one in which small changes in the input lead to small changes in the output, without any abrupt jumps or breaks. This concept is significant in integral calculus because continuity often ensures that functions behave well under integration. Moreover, according to the Intermediate Value Theorem, if a function is continuous on a closed interval \( [a, b] \), it takes on every value between \( f(a) \) and \( f(b) \) at some point within the interval. This theorem is particularly crucial in understanding how we can construct suitable functions in proofs, like the function \( g(x) \) in the original exercise. Ensuring that \( g(x) \) is continuous allows for the application of the zero integral property in certain proofs.

Zero Integral Property

The zero integral property is an important observation in integral calculus, asserting that if the integral of a continuous function over a certain interval is zero, then under certain conditions, this can imply that the function itself must be zero throughout that interval.
\[ \text{If } \int_{a}^{b} f(x)dx = 0 \text{ and f(x) is continuous on } [a, b], \text{ then } f(x) = 0 \text{ for all } x \in [a, b]. \]
This property is crucial to the exercise at hand. The core idea revolves around understanding that, should there be any point in the interval where the function isn't zero, the integral wouldn't be zero either. Conversely, this property underpins many theoretical aspects of integral calculus, providing the basis for more advanced techniques and theorems.

Contradiction Method

The contradiction method, also known as proof by contradiction, is a classic technique in mathematical proofs. This method involves assuming the opposite of what you intend to prove and then demonstrating that this assumption leads to a contradiction - an outcome that is logically impossible. The presence of a contradiction indicates that the initial assumption must be false, affirming the truth of the original statement.
In the context of the original exercise, the contradiction method plays a pivotal role. By assuming that the function \( f(x) \) is not zero everywhere in the interval and then showing that this leads to the violation of the zero integral property, one uncovers a contradiction. This effectively reinforces the truth of the statement we set out to prove: that \( f(x) \) must necessarily be zero everywhere on the interval if the integral of the product of \( f(x) \) and any continuous function \( g(x) \) is zero across that interval.