Which of the following statements are true?
- AIf $f(x)$ is continuous and periodic with period $T$, then $I = \int_a^{a+T} f(x) dx$ depends on $a$.
- BIf $f(x)$ is continuous and periodic with period $T$, then $I = \int_a^{a+T} f(x) dx$ does not depend on $a$.
- CLet $f(x) = \begin{cases} 1, & \text{if } x \in \mathbb{Q} \\ 0, & \text{if } x \notin \mathbb{Q} \end{cases}$, then $f$ is periodic with period $T$ only if $T$ is rational.
- D$f$ defined in $(C)$ is periodic for all $T \in \mathbb{Q} \setminus \{0\}$.
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$I.$ $A$ differentiable function $f$ with maximum at $x = c$ $\implies f''(c) < 0$.
$II.$ Antiderivative of a periodic function is also a periodic function.
$III.$ If $f$ has a period $T$ then for any $a \in R$,$\int\limits_0^T {f(x)\,dx} = \int\limits_0^T {f(x + a)\,dx}$.
$IV.$ If $f(x)$ has a maxima at $x = c$,then $f$ is increasing in $(c - h, c)$ and decreasing in $(c, c + h)$ as $h \to 0$ for $h > 0$. Now indicate the correct alternative.
$I.$ $A$ differentiable function $f$ with maximum at $x = c$ $\implies f''(c) < 0$.
$II.$ Antiderivative of a periodic function is also a periodic function.
$III.$ If $f$ has a period $T$ then for any $a \in R$,$\int\limits_0^T {f(x)\,dx} = \int\limits_0^T {f(x + a)\,dx}$.
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