Which one of the following statements is NOT CORRECT ?

Question

Which one of the following statements is NOT CORRECT ? (A) The derivative of a diffrentiable periodic function is a periodic function with the same period. (B) If f ( x ) and g ( x ) both are defined on the entire number line and are aperiodic then the function F(x)=f(x) ⋅ g(x) can not be periodic. (C) Derivative of an even differentiable function is an odd function and derivative of an odd differentiable function is an even function. (D) Every function f ( x ) can be represented as the sum of an even and an odd function

Tardigrade Admin · Accepted Answer

For (B): consider f(x)=x-√x2+1 and g(x)=x+√x2+1 then F(x)=1 which is periodic ⇒ False

Q. Which one of the following statements is NOT CORRECT ?

The derivative of a diffrentiable periodic function is a periodic function with the same period.

If $f (x)$ and $g (x)$ both are defined on the entire number line and are aperiodic then the function $F (x) = f (x) \cdot g (x)$ can not be periodic.

Derivative of an even differentiable function is an odd function and derivative of an odd differentiable function is an even function.

Every function $f (x)$ can be represented as the sum of an even and an odd function

For $(B)$ : consider $f (x) = x - x^{2} + 1$ and $g (x) = x + x^{2} + 1$ then $F (x) = 1$ which is periodic $\Rightarrow$ False

Q. Which one of the following statements is NOT CORRECT ?

The derivative of a diffrentiable periodic function is a periodic function with the same period.

If f(x) and g(x) both are defined on the entire number line and are aperiodic then the function F(x)=f(x)⋅g(x) can not be periodic.

Derivative of an even differentiable function is an odd function and derivative of an odd differentiable function is an even function.

Every function f(x) can be represented as the sum of an even and an odd function

For (B) : consider f(x)=x−x2+1​ and g(x)=x+x2+1​ then F(x)=1 which is periodic ⇒ False

If $f (x)$ and $g (x)$ both are defined on the entire number line and are aperiodic then the function $F (x) = f (x) \cdot g (x)$ can not be periodic.

Every function $f (x)$ can be represented as the sum of an even and an odd function

For $(B)$ : consider $f (x) = x - x^{2} + 1$ and $g (x) = x + x^{2} + 1$ then $F (x) = 1$ which is periodic $\Rightarrow$ False