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Prove that if $F(x)$ and $G(x)$ are antiderivatives of $f(x)$ , then there exists a constant such that $F(x)-G(x)=C$ .

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Lesson Parent:

Problem:

Answer:

It is true that if $F(x)$ and $G(x)$ are antiderivatives of $f(x)$ , then there exists a constant such that $F(x)-G(x)=C$ .

Solution:

Now since $-1$ is a constant and $G(x)$ is differentiable we know that

Since, both $F(x)$ and $-G(x)$ are differentiable then we know that

and since

we know that

Now let's call <Sign in to see all the formulas>. Because of our properties of function limits, we know that $H(x)$ is continuous on $[a,b]$ . Similarly, from our differentiation formulas we know that <Sign in to see all the formulas> on $(a,b)$ . From this problem we know that $H(x)$ is a constant. Let's call the constant $C$ so that <Sign in to see all the formulas> and therefore, <Sign in to see all the formulas>.

The proof is now complete.