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Graphical Interpretation of Relation between Continuity and Differentiability

Let f1:0, ∞ →...

Question

Let $f_{1} : (0, \infty) \to R$ and $f_{2} : (0, \infty) \to R$ be defined by $f_{1} (x) = \int_{0}^{x} 21 \prod j = 1 (t - j)^{j} d t,$ $x > 0$ and $f_{2} (x) = 98 (x - 1)^{50} - 600 (x - 1)^{49} + 2450,$ $x > 0,$ where, for any positive integer $n$ and real number $a_{1}, a_{2} \dots a_{n},$ $n \prod i = 1 a_{i}$ denotes the product of $a_{1}, a_{2}, . . . a_{n} .$ 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 $(0, \infty) .$

The value of $6 m_{1} + 4 n_{2} + 8 m_{2} n_{2}$ is

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Solution

$f_{2} (x) = 98 (x - 1)^{50} - 600 (x - 1)^{49} + 2450$
$\Rightarrow f_{2}^{'} (x) = 98 \times 50 (x - 1)^{49} - 600 \times 49 \times (x - 1)^{48}$
$= 98 \times 50 \times (x - 1)^{48} (x - 7)$
$f_{2} (x)$ has local minimum at $x = 7$ and no local maximum.
$\Rightarrow m_{2} = 1, n_{2} = 0$
$6 m_{2} + 4 n_{2} + 8 m_{2} n_{2}$
$= 6 \times 1 + 4 \times 0 + 8 \times 1 \times 0 = 6$

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