f-x-9-x-m-3-x-m-6-x-R-f-x-f-x-0-find-the-range-of-m-

Question Number 182012 by CrispyXYZ last updated on 03/Dec/22

f (x) = 9^{x} - m \cdot 3^{x} + m + 6

\exists x \in R, f (x) + f (- x) = 0

find the range of m .

Answered by mr W last updated on 04/Dec/22

f (x) + f (- x) = 0

9^{x} + 9^{- x} - m (3^{x} + 3^{- x}) + 2 (m + 6) = 0

{(3^{x} + 3^{- x})}^{2} - m (3^{x} + 3^{- x}) + 2 (m + 5) = 0

3^{x} + 3^{- x} = \frac{m \pm \sqrt{{(m - 4)}^{2} - 56}}{2} ⩾ 2

\Rightarrow m \pm \sqrt{{(m - 4)}^{2} - 56} ⩾ 4

m - 4 ⩾ \pm \sqrt{{(m - 4)}^{2} - 56}

\Rightarrow m - 4 ⩾ \sqrt{56} = 2 \sqrt{14} \Rightarrow m ⩾ 4 + 2 \sqrt{14}

a n s w e r i s m \in [4 + 2 \sqrt{14}, + \infty)

Commented by manxsol last updated on 03/Dec/22

a + b ≫ 2 \sqrt{a b}

3^{x} + 3^{- x} ≫ 2 \sqrt{3^{x} . 3^{- x}} = 2

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