Question

If $F$ and $f$ are differentiable functions such that $F ( x )=\int\limits_0^{ x } f ( t ) dt$, and if $F ( a )=-2$ and $F ( b )=-2$ where $a < b$, then which of the following must be true?

• A. $f ( x )=0$ for some $x$ such that $a < x < b$.
• B. $f ( x )<0$ for all $x$ such that $a < x < b$.
• C. $F ( x ) \leq 0$ for all $x$ such that $a < x < b$.
• D. $F ( x )=0$ for some $x$ such that $a < x < b$.

Answer 1

Answer: (A)

Use Rolle's Theorem for $F ( x )$ in $[ a , b ] ; F ( a )= F ( b )$, also $f$ is differentiable
Hence, we have $F^{\prime}(x)=f(x)=0 \Rightarrow$ continuous
[12th, 26-08-2012, P-1] for some $x \in(a, b) \Rightarrow A$