Let $f:\lbrack 1,\infty) \rightarrow \mathbb{R}$ be a differentiable function, If$6\int_{1}^{x}\mspa

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Let $f:\lbrack 1,\infty) \rightarrow \mathbb{R}$ be a differentiable function, If$6\int_{1}^{x}\mspace{2mu} f(t)dt = 3xf(x) + x^{3} - 4$ for all $x \geq 1$, then the value of $f(2) - f(3)$ is

Accepted Answer

$\ 6\int_{1}^{x}\mspace{2mu} f(t)dt = 3xf(x) + x^{3} - 4$

Diff. both side

$${6f(x) = 3xf'(x) + 3f(x) + 3x^{2} }{3f(x) = 3xf'(x) + 3x^{2} }{x\frac{dy}{dx} - y = - x^{2} }{\frac{x\frac{dy}{dx} - 9}{x^{2}} = - 1 }{\Rightarrow \frac{d}{dx}\left( \frac{y}{x} \right) = - 1 }{\frac{y}{x} = - x + C }{\Rightarrow f(x) = - x^{2} + Cx }$$at $x = 1,y = 1 \Rightarrow C = 2$

$${f(x) = - x^{2} + 2x }{f(2) - f(3) = 3}$$

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