Which one of $\mathbf { A } - \mathbf { F }$ correctly completes the following statement? Given that $a < b$, and $\mathrm { f } ( x ) > 0$ for all $x$ with $a < x < b$, the trapezium rule produces an overestimate for $\int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { d } x \ldots$ A ... if $\mathrm { f } ^ { \prime } ( x ) > 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) < 0$ for all $x$ with $a < x < b$ B ... only if $\mathrm { f } ^ { \prime } ( x ) > 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) < 0$ for all $x$ with $a < x < b$ C ... if and only if $\mathrm { f } ^ { \prime } ( x ) > 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) < 0$ for all $x$ with $a < x < b$ D ... if $\mathrm { f } ^ { \prime } ( x ) < 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) > 0$ for all $x$ with $a < x < b$ E $\ldots$ only if $\mathrm { f } ^ { \prime } ( x ) < 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) > 0$ for all $x$ with $a < x < b$ F ... if and only if $\mathrm { f } ^ { \prime } ( x ) < 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) > 0$ for all $x$ with $a < x < b$
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Which one of $\mathbf { A } - \mathbf { F }$ correctly completes the following statement?\\
Given that $a < b$, and $\mathrm { f } ( x ) > 0$ for all $x$ with $a < x < b$, the trapezium rule produces an overestimate for $\int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { d } x \ldots$
A ... if $\mathrm { f } ^ { \prime } ( x ) > 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) < 0$ for all $x$ with $a < x < b$\\
B ... only if $\mathrm { f } ^ { \prime } ( x ) > 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) < 0$ for all $x$ with $a < x < b$\\
C ... if and only if $\mathrm { f } ^ { \prime } ( x ) > 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) < 0$ for all $x$ with $a < x < b$\\
D ... if $\mathrm { f } ^ { \prime } ( x ) < 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) > 0$ for all $x$ with $a < x < b$\\
E $\ldots$ only if $\mathrm { f } ^ { \prime } ( x ) < 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) > 0$ for all $x$ with $a < x < b$\\
F ... if and only if $\mathrm { f } ^ { \prime } ( x ) < 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) > 0$ for all $x$ with $a < x < b$