Which of the following statements are true?

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B, D) For a periodic function $f(x)$ with period $T$, the integral over any interval of length $T$ is constant, i.e.,$\int_a^{a+T} f(x) dx = \int_0^T

Vedclass · Accepted Answer

$f$ defined in $(C)$ is periodic for all $T \in \mathbb{Q} \setminus \{0\}$.

Vedclass · Answer

(B, D) For a periodic function $f(x)$ with period $T$, the integral over any interval of length $T$ is constant, i.e.,$\int_a^{a+T} f(x) dx = \int_0^T f(x) dx$. Thus, statement $(B)$ is true and $(A)$ is false.For the Dirichlet function $f(x) = \begin{cases} 1, & x \in \mathbb{Q} \ 0, & x 
otin \mathbb{Q} \end{cases}$, $f(x+T) = f(x)$ holds for any rational $T$ because if $x$ is rational, $x+T$ is rational, and if $x$ is irrational, $x+T$ is irrational. Thus, $f$ is periodic for all rational $T$. Statement $(D)$ is true.

Similar Questions

Let $f$ and $g$ be continuous functions on $[0, a]$ such that $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$,then $\int_0^a f(x) g(x) d x$ is equal to

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be continuous functions. Then the value of the integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} [f(x)+f(-x)][g(x)-g(-x)] \, dx$ is

The value of $\int_{-7}^{7} \frac{5^x}{5^{[x]}} dx$ is equal to (where $[.]$ denotes the greatest integer function).

If $f(x) = \cos(\tan^{-1}x)$,then the value of the integral $\int_{0}^{1} x f''(x) dx$ is

$\int_{0}^{1} (1 + |\sin x|)(ax^2 + bx + c) dx = \int_{0}^{2} (1 + |\sin x|)(ax^2 + bx + c) dx$. Then,the location of the roots of $ax^2 + bx + c = 0$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module