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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?

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    I've put the problem statement in LaTeX- please check I did not change the meaning of what you wrote. What does $f(x)\frac{d}{dx}$ mean? Do you want it to mean the derivative of $f(x)$ with respect to $x$? This is more commonly denoted $\frac{d}{dx}f(x)$ or $\frac{df(x)}{dx}$.2012-08-06
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    Apart from notational issues, your calculations are correct.2012-08-06
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    yes i meant the derivative of f(x) with respect to x2012-08-06

2 Answers 2