The following is an exercise from Spivak's Calculus:
Find all continuous functions $f$ satisfying $$\int_0^xf(t)dt=(f(x))^2+C$$ for $C\neq 0$, assuming that $f$ has at most one zero.
I have several questions before bringing up the proof:
- What is significant about $C\neq 0$?
- What is significant about $f$ being continuous?
- What is significant about $f$ having at most one root?
Here is my proposed solution: (Which according to the answer key, is incorrect) By FTC, we know that $(f(x)))^2$ is a differentiable function, thus $(f(x))^2+C$ is also differentiable. Then, \begin{align*} \frac{d}{dx}\int_0^xf(t)dt & = \frac{d}{dx}\Big[(f(x))^2+C\Big] \\ f(x) & = 2f(x)f'(x) \\ 2f(x)f'(x)-f(x) & =0 \\ f(x)(2f'(x)-1) & =0 \end{align*} So either $f(x)=0$ or $f'(x)=1/2$. The problem states that $f(x)=0$ for only one point, so $f'(x)=1/2$ everywhere else. Since $f=\int_0^xf'(t)dt=\int_0^x(1/2)dt=x/2+C$.
However, the answer key claims that $f$ is constant? I'm pretty confused, and I didn't use all the facts that the problem gave.
First of all, note that the $C$ in your answer has nothing to do with the $C$ from the question. Furthermore, where did you read “assuming that $f$ has at most one zero”? I am looking at the third edition of Spivak's Calculus and I don't see it there.
Anyway, you were right when you got that $f(x)=\frac x2$. But then$$\int_0^x f(t)\,\mathrm dt-\bigl(f(x)\bigr)^2=\frac{x^2}4-\frac{x^2}4=0,$$and $C$ is supposed to be different from $0$.