Let \( f_{1}:(0, \infty) \rightarrow R \) and \( f_{2}:(0, \infty) \rightarrow R \) be defined b...

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Let \( f_{1}:(0, \infty) \rightarrow R \) and \( f_{2}:(0, \infty) \rightarrow R \) be defined by
\[
f_{1}(x)=\int_{0}^{x} \prod_{j=1}^{21}(t-j)^{j} d t, x0
\]
and \( f_{2}(x)=98(x-1)^{50}-600(x-1)^{49}+2450, x0 \), where for any positive integer \( n \) and real numbers \( a_{1}, a_{2}, \ldots a_{n}, \prod_{i=1}^{n} a_{i} \) denotes the product of \( a_{1}, a_{2}, \ldots, a_{n} \). Let \( m_{i} \) and \( n_{i} \), respectively, denote the number of points of local minima and the number of points of local maxima of function \( f_{i}, i=1,2 \), in the interval \( (0, \infty) \)
Consider the equation
\[
\int_{1}^{e} \frac{\left(\log _{e} x\right)^{1 / 2}}{x\left(a-\left(\log _{e} x\right)^{3 / 2}\right)^{2}} d x=1, \quad a \in(-\infty, 0) \cup(1, \infty)
\]
Which of the following statements is/are TRUE?
(a) No \( a \) satisfies the above equation
(b) An integer \( a \) satisfies the abvoe equation
(c) An irrational number \( a \) satisifies the abvoe equation.
(d) More than one \( a \) satisfy the above equation
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