Problem:
(a). If $f$ is continuous on $[a,b]$ and $\int_a^x f(t) dt = 0$ for all $x \in [a,b]$, show that $f(x) = 0$ for all $x \in [a,b]$.
(b). If $f$ is continuous on $[a,b]$ and $\int_a^x f(t)dt = \int_x^b f(t)dt$ for all $x \in [a,b]$, show that $f(x)=0$ for all $x\in [a,b]$.
Work so far:
For (a), I think I am supposed to use Leibniz's rule and differentiate both sides and say $f(x)d/dx(x) - f(a)d/dx(a) = 0,$ so $f(x)-0=0$ and $f(x)=0.$ For (b) I think I am supposed to use Leibniz's Rule and differentiate both sides and get $f(x)d/dx(x) - f(a)d/dx(a) = f(b)d/dx(b) - f(x)d/dx(x)$, thus $f(x) - 0 = 0 - f(x)$, $2f(x) = 0$, and $f(x) = 0$....am I going about this correctly?