Which one of the following statements is NOT | vedclass

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(B) Let us analyze each statement:$A$: If $f(x)$ is periodic with period $T$,then $f'(x) = \lim_{h 	o 0} \frac{f(x+h)-f(x)}{h}$. Since $f(x+T+h) = f(

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

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(B) Let us analyze each statement:$A$: If $f(x)$ is periodic with period $T$,then $f'(x) = \lim_{h 	o 0} \frac{f(x+h)-f(x)}{h}$. Since $f(x+T+h) = f(x+h)$ and $f(x+T) = f(x)$,$f'(x+T) = f'(x)$. Thus,the derivative is periodic with the same period. This is $CORRECT$.$B$: Consider $f(x) = x - \sqrt{x^2+1}$ and $g(x) = x + \sqrt{x^2+1}$. Both are aperiodic. However,$F(x) = f(x) \cdot g(x) = (x^2 - (x^2+1)) = -1$. The constant function $-1$ is periodic (with any period $T > 0$). Thus,the statement that it cannot be periodic is $FALSE$.$C$: If $f(x)$ is even,$f(-x) = f(x)$. Differentiating gives $-f'(-x) = f'(x)$,so $f'(-x) = -f'(x)$ (odd). If $f(x)$ is odd,$f(-x) = -f(x)$. Differentiating gives $-f'(-x) = -f'(x)$,so $f'(-x) = f'(x)$ (even). This is $CORRECT$.$D$: Any function $f(x)$ can be written as $f(x) = \frac{f(x)+f(-x)}{2} + \frac{f(x)-f(-x)}{2}$,where the first part is even and the second is odd. This is $CORRECT$.Therefore,the incorrect statement is $B$.

Which one of the following statements is $NOT \text{ } CORRECT$?

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