MCQ
The minimum value of $f(a) = (2{a^2} - 3) + 3(3 - a) + 4$ is
- A${{15} \over 2}$
- B${{11} \over 2}$
- C${{ - 13} \over 2}$
- ✓${{71} \over 8}$
$f'(a) = 4a - 3,f(a) = 4 > 0$
for exteremum, $f'(a) = 0 \Rightarrow a = \frac{3}{4}$
$\therefore$ $f(a)$ is minimum at $a = \frac{3}{4}$
$f{(a)_{\min }} = 2 \times {\left( {\frac{3}{4}} \right)^2} - 3 \times \left( {\frac{3}{4}} \right) + 10 = \frac{{71}}{8}$.
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Then for the objective function $z=-x+2 y$
$(i)$ Maximum value of $z$ has at $\ldots \ldots \ldots . . .$
$(ii)$ Minimum value of $z$ has at $\ldots \ldots \ldots . . .$
$(iii)$ The maximum value of $z$ is $\ldots \ldots \ldots . . .$
$(iv)$ The minimum value of $z$ is $\ldots \ldots \ldots . . .$
$f_1(x)=\int_0^x \prod_{j=1}^{21}( t - j )^{ j } dt , x >0$
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 ($2$)