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Question Number 31670 by gunawan last updated on 12/Mar/18

how many roots from equation

{a e}^{x} = 1 + x + \frac{x^{2}}{2}

f r o m a > 0 ?

Answered by mrW2 last updated on 12/Mar/18

f (x) = {a e}^{x} (a > 0)

g (x) = 1 + x + \frac{x^{2}}{2}

w i t h x \to - \infty :

f (x) \to 0, g (x) \to + \infty

i . e . f (x) < g (x)

w i t h x \to + \infty :

f (x) \to + \infty, g (x) \to + \infty

\frac{f (x)}{g (x)} \to + \infty

i . e . f (x) > g (x)

\Rightarrow b e t w e e n - \infty a n d + \infty t h e r e i s o n e

i n t e r s e c t i o n p o i n t f r o m f (x) a n d g (x),

i . e . {a e}^{x} = 1 + x + \frac{x^{2}}{2} h a s a l w a y s o n e a n d

o n l y o n e s o l u t i o n .

Commented by gunawan last updated on 12/Mar/18

Thank you very much

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