If f : R to R is a continuous function such t | vedclass

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(C) Given the equation $f(x) = \int\limits_1^x {tf(t)dt}$.By the Fundamental Theorem of Calculus,differentiating both sides with respect to $x$,we get

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$\int\limits_{ - 3}^3 {f(x)dx = 0}$

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(C) Given the equation $f(x) = \int\limits_1^x {tf(t)dt}$.By the Fundamental Theorem of Calculus,differentiating both sides with respect to $x$,we get $f'(x) = xf(x)$.Also,substituting $x = 1$,we get $f(1) = \int\limits_1^1 {tf(t)dt} = 0$.Now,$\frac{f'(x)}{f(x)} = x$.Integrating both sides with respect to $x$,we get $\ln|f(x)| = \frac{x^2}{2} + C$.This implies $f(x) = A e^{x^2/2}$.Using the condition $f(1) = 0$,we have $A e^{1/2} = 0$,which implies $A = 0$.Therefore,$f(x) = 0$ for all $x \in R$.Since $f(x) = 0$,the integral $\int\limits_{-3}^3 f(x) dx = \int\limits_{-3}^3 0 dx = 0$.Thus,the correct statement is $\int\limits_{-3}^3 f(x) dx = 0$.

If $f : R \to R$ is a continuous function such that $f(x) = \int\limits_1^x {tf(t)dt}$,then the correct statement is -

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