The function $\mathrm { f } ( x )$ is defined for all real values of $x$. Which of the following conditions on $\mathrm { f } ( x )$ is/are necessary to ensure that $$\int _ { - 5 } ^ { 0 } \mathrm { f } ( x ) \mathrm { d } x = \int _ { 0 } ^ { 5 } \mathrm { f } ( x ) \mathrm { d } x$$ Condition I: $\quad \mathrm { f } ( x ) = \mathrm { f } ( - x ) $ for $- 5 \leq x \leq 5$ Condition II: $\mathrm { f } ( x ) = c$ for $- 5 \leq x \leq 5$, where $c$ is a constant Condition III: $\mathrm { f } ( x ) = - \mathrm { f } ( - x ) $ for $- 5 \leq x \leq 5$ A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III
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The function $\mathrm { f } ( x )$ is defined for all real values of $x$.\\
Which of the following conditions on $\mathrm { f } ( x )$ is/are necessary to ensure that
$$\int _ { - 5 } ^ { 0 } \mathrm { f } ( x ) \mathrm { d } x = \int _ { 0 } ^ { 5 } \mathrm { f } ( x ) \mathrm { d } x$$
Condition I: $\quad \mathrm { f } ( x ) = \mathrm { f } ( - x ) $ for $- 5 \leq x \leq 5$\\
Condition II: $\mathrm { f } ( x ) = c$ for $- 5 \leq x \leq 5$, where $c$ is a constant\\
Condition III: $\mathrm { f } ( x ) = - \mathrm { f } ( - x ) $ for $- 5 \leq x \leq 5$
A none of them\\
B I only\\
C II only\\
D III only\\
E I and II only\\
F I and III only\\
G II and III only\\
H I, II and III