if-a-b-c-1-then-prove-that-a-1-a-b-1-b-c-1-c-are-the-sides-of-a-triangle-

Question Number 155481 by mathdanisur last updated on 01/Oct/21

if a; b; c \in [1; \infty)

then prove that a^{\frac{1}{a}}; b^{\frac{1}{b}}; c^{\frac{1}{c}}

are the sides of a triangle .

Answered by mr W last updated on 01/Oct/21

f o r x \in [1, \infty) :

1 ⩽ x^{\frac{1}{x}} ⩽ e^{\frac{1}{e}} \approx 1.4447

t h a t m e a n s 1 ⩽ a^{\frac{1}{a}}, b^{\frac{1}{b}}, c^{\frac{1}{c}} ⩽ e^{\frac{1}{e}} .

s a y a^{\frac{1}{a}} ⩽ b^{\frac{1}{b}} ⩽ c^{\frac{1}{c}},

c^{\frac{1}{c}} ⩽ e^{\frac{1}{e}} \approx 1.4447

a^{\frac{1}{a}} + b^{\frac{1}{b}} ⩾ 1 + 1 = 2

\Rightarrow a^{\frac{1}{a}} + b^{\frac{1}{b}} > c^{\frac{1}{c}}

i . e . a^{\frac{1}{a}}, b^{\frac{1}{b}}, c^{\frac{1}{c}} c a n f o r m a t r i a n g l e .

Commented by mathdanisur last updated on 01/Oct/21

Very nice solution, thank you S er