1-prove-the-convexity-of-f-x-ln-1-e-x-2-prove-that-x-1-x-2-x-n-R-n-1-k-1-n-x-k-1-n-k-1-n-x-k-1-1-n-3-prove-that-1-n-1-n-n-1-1-n-

Question Number 51984 by maxmathsup by imad last updated on 01/Jan/19

1) p r o v e t h e c o n v e x i t y o f f (x) = l n (1 + e^{x})

2) p r o v e t h a t \forall (x_{1}, x_{2}, \dots, x_{n}) \in R^{n}

1 + \prod_{k = 1}^{n} {(x_{k})}^{\frac{1}{n}} ⩽ \prod_{k = 1}^{n} {(x_{k} + 1)}^{\frac{1}{n}}

3) p r o v e t h a t 1 + {(n!)}^{\frac{1}{n}} ⩽ {((n + 1)!)}^{\frac{1}{n}}