#### Consider $f(x)=\int_1^x\left(t+\frac{1}{t}\right) d t$and $g(x)=f^{\prime}(x) for x \in\left[\frac{1}{2}, 3\right].$ If P is a point on the curve $y=g(x)$ such that the tangent to this curve at P is parallel to a chord joining the points $\left(\frac{1}{2}, g\left(\frac{1}{2}\right)\right)$ and $(3, g(3))$ of the curve, then the coordinates of the point P Option: 1  can't be found outOption: 2 $\left(\frac{7}{4}, \frac{65}{28}\right)$Option: 3  (1,2)Option: 4 $\left(\sqrt{\frac{3}{2}}, \frac{5}{\sqrt{6}}\right)$

$f(x)=\int_1^x\left(t+\frac{1}{t}\right) d t \Rightarrow f^{\prime}(x)=x+\frac{1}{x} \\$

\begin{aligned} \begin{array}{ccc} \therefore & g(x)=x+\frac{1}{x} \text { for } x \in\left[\frac{1}{2}, 3\right] \\ & g\left(\frac{1}{2}\right)=2+\frac{1}{2}=\frac{5}{2}, g(3)=3+\frac{1}{3}=\frac{10}{3}\\ Let &P \equiv(c, g(c)), c \in\left[\frac{1}{2}, 3\right] \end{array} \end{aligned}
\begin{aligned} \begin{array}{ccc} \therefore & g(x)=x+\frac{1}{x} \text { for } x \in\left[\frac{1}{2}, 3\right] \\ & g\left(\frac{1}{2}\right)=2+\frac{1}{2}=\frac{5}{2}, g(3)=3+\frac{1}{3}=\frac{10}{3}\\ Let &P \equiv(c, g(c)), c \in\left[\frac{1}{2}, 3\right] \\ \text{By LMVT,} & g^{\prime}(c)=\frac{g(3)-g\left(\frac{1}{2}\right)}{3-1 / 2} \\ \therefore & 1-\frac{1}{c^2}=\frac{\frac{10}{3}-\frac{5}{2}}{3-\frac{1}{2}} \end{array} \end{aligned}

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