x-1-1-x-gt-e-1-e-1-Solve-for-x-x-N-2-Solve-for-x-x-gt-0-

Question Number 182575 by CrispyXYZ last updated on 11/Dec/22

\sqrt[x]{x + 1} > \sqrt[e]{e}

1) Solve for x (x \in N^{+}) .

2) Solve for x (x > 0) .

Answered by mr W last updated on 12/Dec/22

f (x) = {(1 + x)}^{\frac{1}{x}}

f o r x > 0 :

f (x) i s s t r i c t l y d e c r e a s i n g .

f (x) = {(1 + x)}^{\frac{1}{x}} = e^{\frac{1}{e}} \Rightarrow x_{1} \approx 4.7591 (s o l u t i o n s e e l a t e r)

{(1 + x)}^{\frac{1}{x}} > e^{\frac{1}{e}}

\Rightarrow 0 < x < x_{1} \approx 4.7591

1) x \in [1, 4]

2) x \in (0, 4.7591)

Commented by mr W last updated on 12/Dec/22

h o w t o s o l v e {(x + 1)}^{\frac{1}{x}} = a w i t h a > 0

(x + 1) = a^{x}

a (x + 1) = a^{x + 1} = e^{(x + 1) \ln a}

- \ln a (x + 1) e^{- (x + 1) \ln a} = - \frac{\ln a}{a}

- \ln a (x + 1) = W (- \frac{\ln a}{a})

\Rightarrow x = - \frac{W (- \frac{\ln a}{a})}{\ln a} - 1

w i t h a = e^{\frac{1}{e}},

x = - e W (- \frac{1}{e^{1 + \frac{1}{e}}}) - 1

\approx - e \times (- 2.118666) - 1 = 4.7591 o r

\approx - e \times (- 0.367879) - 1 = - 1.1992 (r e j e c t e d)

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