Which of the following is/are CORRECT?

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If $x_{1}, x_{1}, x_{2}, x_{3}, ........ x_{n - 1}$ be n zero’s of the polynomial $P (x) = x^{n} + α x + β$ , where $x_{i} \neq x_{j} \forall i, j$ then $(x_{1} - x_{2}) (x_{1} - x_{3}) (x_{1} - x_{4}) ...... (x_{1} - x_{n - 1}),$ equals $n (n - 1) x_{1}^{n - 2}$ .

Let f(0) = 0 and $\int_{0}^{2} f' (2 t) e^{f (2 t)} d t = 5$ , then the value of f(4) equals ln(11).

The minimum value of the function $f (x) = x \sqrt{x} + \frac{1}{x \sqrt{x}} - 4 (x + \frac{1}{x})$ for all permissible real x, is –10.

Let f(x) be a differentiable function such that $f (x) + f' (x) \leq 1 \forall x \in R$ and f(0) = 0, then the greatest value of f(1) is $1 - \frac{1}{e}$
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answer is A, B, D.

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Detailed Solution

A) $x = t^{2} f (t) = t^{3} + \frac{1}{t^{3}} - 4 (t^{2} + \frac{1}{t^{2}}) f^{/} (t) = 3 t^{2} - \frac{3}{t^{4}} - 8 t + \frac{8}{t^{3}} = \frac{3 t^{6} - 8 t^{5} + 8 t - 3}{t^{4}} t = 1, - 1, \frac{3 - \sqrt{5}}{2}, \frac{3 + \sqrt{5}}{2} f_{m i n i m u m} = {(t + \frac{1}{t})}^{3} - 3 (t + \frac{1}{t}) - 4 {(t + \frac{1}{t})}^{2} + 8$

= 27 – 9 – 36 + 8
$= - 10 f o r t + \frac{1}{t} = 3$
B) y = 2t
$_{0}^{4} \frac{f^{^{/}} (y) e^{f (y)} d y}{2} = 5$
f(4) = ln11
C) $x^{n} + α x + β = (x - x_{1}) (x - x_{1}) (x - x_{2}) (x - x_{3}) - - - - (x - x_{n - 1})$
Differentiate with respect to x twice and substitute
$n (n - 1) {x_{1}}^{n - 2} = 2 (x_{1} - x_{2}) (x_{1} - x_{3}) - - - - (x_{1} - x_{n - 1})$
D) $f (x) + f^{/} (x) \leq 1 e^{x} f (x) - e^{0} f (0) \leq e^{x} - 1$