MCQ
${d \over {dx}}\{ {e^{ - a{x^2}}}\log (\sin x)\} = $
- A${e^{ - a{x^2}}}(\cot x + 2ax\log \sin x)$
- B${e^{ - a{x^2}}}(\cot x + ax\log \sin x)$
- ✓${e^{ - a{x^2}}}(\cot x - 2ax\log \sin x)$
- DNone of these
$ = {e^{ - a{x^2}}}( - 2ax).\log (\sin x) + {e^{ - a{x^2}}}\cot x$
$ = {e^{ - ax}}^2[\cot x - 2ax\log (\sin x)]$.
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$f_1(x)=\int_0^x \prod_{j=1}^{21}( t - j )^{ j } dt , x >0$
and
$f_2(x)=98(x-1)^{50}-600(x-1)^{39}+2450, x>0,$
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 $(0, \infty)$
($2$) The value of $2 m_1+3 n_1+m_1 n_1$ is. . . . . .
($2$) The value of $6 m _2+4 n _2+8 m _2 n _2$ is. . . . . .
Give the answer or quetion ($1$) and ($2$)