t-1-F-t-2-pi-0-pi-2-1-tcos-2-d-1-Show-that-t-1-F-t-1-t-F-1-1-t-2-show-that-if-0-t-1-0-F-t-2-F-t-1-t-2-t-1-4-

Question Number 146697 by ArielVyny last updated on 15/Jul/21

\forall t ⩾ - 1, F (t) = \frac{2}{π} \int_{0}^{\frac{π}{2}} \sqrt{1 + {t c o s}^{2} φ} d φ

1) S h o w t h a t \forall t ⩽ - 1 F (t) = \sqrt{1 + t} F (- \frac{1}{1 + t})

2) s h o w t h a t i f 0 ⩽ t_{1},

0 ⩽ F (t_{2}) - F (t_{1}) ⩽ \frac{t_{2} - t_{1}}{4}