First slide

Maxima and minima

Question

$Let f (x) = (1 + x)^{n} - (1 + n x), x \in [- 1, \infty), then f$

Moderate

A

has an absolute maximum at x = 0
B

has neither absolute maximum nor absolute minimum at x = 0
C

has an absolute minimum at x = 0
D

does not have absolute minimum at x = 0

Solution

$\begin{array}{l} f (x) = (1 + x)^{n} - (1 + n x) \\ \Rightarrow f^{'} (x) = n [(1 + x)^{n - 1} - 1] . Now for x = 0, f^{'} (0) = 0 and for \end{array}$
$\begin{array}{l} x > 0, f^{'} (x) > 0 . \underline{Thus} f increases on [0, \infty) i.e. f (x) \geq f (0) \\ for x \in [0, \infty) . For - 1 \leq x < 0, 0 \leq 1 + x < 1 \end{array}$
$so (1 + x)^{n} - 1 < 0 . So f decreases on$
$[-1. 0) i.e. f (x) \geq f (0) = 0 for x \in [- 1, 0) . Hence f has an$ $absolute minimum at x = 0$

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