Prove that following functions are increasing in given intervals.
$f (x) = (x - 1) e^{x} + 1, x > 0$ .

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Solution

$\begin{matrix} f (x) = (x - 1) e^{x} + 1, x > 0 d i f f e r e n t i a t e w . r t o x f^{'} (x) (x - 1) \frac{d e^{x}}{d x} + \frac{e^{x} d (x - 1)}{d x} + \frac{d_{1}}{d x} f^{'} (x) = (x - 1) e^{x} + e^{x} + 1 + 0 f^{'} (x) e^{x} (x - 1 + 1) f^{'} (x) = x e^{x} f o r i n c r e a s i n g x e^{x} ⩾ 0 x ⩾ 0 e^{x} ⩾ 0 I t i s i n c r e a s i n g (0, \infty) A n s . \end{matrix}$

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