Let `f(x)` and `phi(x)` are two continuous function on `R` satisfying `phi(x)=int_(a)^(x)f(t)dt, a!=0` and another continuous function `g(x)` satisfyi

Question

Let `f(x)` and `phi(x)` are two continuous function on `R` satisfying `phi(x)=int_(a)^(x)f(t)dt, a!=0` and another continuous function `g(x)` satisfyi

Let `f(x)` and `phi(x)` are two continuous function on `R` satisfying `phi(x)=int_(a)^(x)f(t)dt, a!=0` and another continuous function `g(x)` satisfying `g(x+alpha)+g(x)=0AA x epsilonR, alpha gt0`, and `int_(b)^(2k)g(t)dt` is independent of `b`
If `f(x)` is an odd function, then
A. `phi(x)` is also an odd function
B. `phi(x)` is an even function
C. `phi(x)` is neither an even nor an odd function
D. for `phi(x)` to be an even function, it must satisfy `int_(0)^(a)f(x)dx=0`

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Salim Ahmed Kawsar

Correct Answer - B
`f(x)` is an odd function. Thus, `f(x)=-f(-x)`
`phi(-x)=int_(0)^(-x)f(t)dt`
Put `t=-y`
`:. phi(-x)=int_(-a)^(x)f(-t)(-dt)`
`=int_(-a)^(x)f(t)dt=int_(-a)^(a)f(t)dt+int_(a)^(x)f(t)dt`
`=0+int_(a)^(x)f(t)dt=phi(x)`.

4 views Answered 2023-02-05 15:29