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  • If <img alt="F(x)=\frac{1}{x^2} \int_4^x\left\{4 t^2-2 F^{\prime}(t)\right\} d t" src="https://entrancecorner.oncodecogs.com/gif.latex?F%28x%29%3D%5Cfrac%7B1%7D%7Bx%5E2%7D%20%5Cint_4%5Ex%5Clef

If F(x)=\frac{1}{x^2} \int_4^x\left\{4 t^2-2 F^{\prime}(t)\right\} d t, then F^{\prime}(4)  equals 

 

Option: 1

\frac{32}{9}


Option: 2

\frac{64}{3}


Option: 3

\frac{64}{9}


Option: 4

none of these 


Answers (1)

best_answer

\begin{aligned} & \because F(x)=\frac{1}{x^2} \int_4^x\left\{4 t^2-2 F^{\prime}(t)\right\} d t \\ \\& \text { or } \quad x^2 F(x)=\int_4^x\left\{4 t^2-2 F^{\prime}(t)\right\} d t \end{aligned}

Differentiating both sides w.r.t. x, then 

x^2 F^{\prime}(x)+F(x) \cdot 2 x=4 x^2-2 F^{\prime}(x)

Put, 

        \begin{aligned} 16 F^{\prime}(4)+8 F(4) & =64-2 F^{\prime}(4) \\ \\18 F^{\prime}(4)+0 & =64 \quad[\because F(4)=0, \text { from Eq. (i) }] \\ \\F^{\prime}(4) & =\frac{32}{9} \end{aligned}

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mansi

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