Mathematics

Problem

For a function

f

that is continuous on an open interval

I

, let

F

and

G

satisfy

F'(x)=G'(x)=f(x)

for all

x

I

. Which of the following must be true on the interval

I

? a)

\dfrac{F(x)}{G(x)}

is constant b)

F(x)-G(x) \text{ varies linearly with } x

F(x)-G(x)

is constant d)

F(x)\,G(x)

is constant

Intuitive explanation

The key insight is that if two functions have the same derivative everywhere on an interval, their difference must be constant. This follows directly from the Mean Value Theorem: if $F'(x) = G'(x)$ , then $(F - G)'(x) = 0$ throughout $I$ , which means $F$ and $G$ differ by a fixed constant. This is the fundamental relationship between antiderivatives—any two antiderivatives of the same function differ only by a constant.

Intuitive explanation

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