Vtgyjfy

1 $begingroup$ 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 $Cneq 0$ , assuming that $f$ has at most one zero. I have several questions before bringing up the proof: What is significant about $Cneq 0$ ? 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+CBig] \ f(x) & = 2f(x)f'(x) \ 2f(x)f'(x)-f(x) & =0 \ f(x)(2f'(x)-1) & =0 end{align*} So either $f(x)=0$ ...