Mathematics

Problem

Let

f

be a continuous, positive function on

\mathbb{R}

. For each real number

k

, define

F_k(x)=k+\displaystyle\int_{0}^{x} f(t)\,dt

. Which statement about the graphs of

F_{1}

and

F_{5}

is true? a) The two graphs are parallel and intersect infinitely often because they share the same slope everywhere. b) The graphs intersect exactly once for some

x<0

, where the negative accumulated area offsets their initial 4-unit separation. c) The graphs never intersect because

F_5(x)=F_1(x)+4

for every

x

. d) The graphs intersect only at points where

f(x)=0

, since the integral’s contribution momentarily stops there.

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