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If $f(x)$ is a continuous function such that $\int_{a}^{\infty}f(x)\,dx$ diverges and $a < b$ , then $\int_{b}^{\infty}f(x)\,dx$ also diverges. $\newline$ True $\newline$ False $\newline$

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Identify discrete and continuous random variables

Full solution

Q. If

f(x)

is a continuous function such that

\int_{a}^{\infty}f(x)\,dx

diverges and

a < b

, then

\int_{b}^{\infty}f(x)\,dx

also diverges.

\newline

True

\newline

False

\newline

Properties of Integrals: Let's consider the properties of integrals and continuous functions. $\newline$ If the integral of a continuous function $f(x)$ from $a$ to infinity diverges, it means that the area under the curve from $a$ to infinity is infinite. $\newline$ Since $b$ is greater than $a$ ( $a < b$ ), the integral from $b$ to infinity would be a part of the integral from $a$ to infinity.
Analysis of Divergence: Now, let's analyze the implication of the divergence of the integral from $a$ to infinity. If the integral from $a$ to infinity diverges, then adding or removing a finite part of the integral (from $a$ to $b$ ) should not affect the divergence. This is because subtracting a finite value from infinity still leaves us with infinity.
Implication of Divergence: Therefore, if the integral of $f(x)$ from $a$ to infinity diverges, the integral from $b$ to infinity, which is a subset of the original integral, must also diverge. $\newline$ This is because the function is continuous, and the divergence from $a$ to infinity indicates that the function does not settle into a pattern that would allow the integral to converge from any point onward.

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