With the help of Lagrange's formula, prove that `(a-b)/(a) le log (a/b) le (a-b)/(b)`, for the condition `0 lt b le a.`

In a triangle ABC, prove that (b + c)/(a) le cosec.(A)/(2)

Using Lagranges mean value theorem,prove that (b-a)/(b)

Prove that |vec(a)|-|vec(b)|le |vec(a)-vec(b)| .

Using Lagranges mean value theorem,prove that (b-a)/(b)

For a , b in R prove that (|a+b|)/(1+|a+b|) le (|a|)/(1+|a|)+(|b|)/(1+|b|)

Using Lagrange's mean value theorem prove that if b gt a gt 0 "then " (b-a)/(1+b^(2)) lt tan^(-1) b -tan^(-1) a lt (b-a)/(1+a^(2))

If P(E) denotes the probability of an event E then (a) 0 lt P(E) le 1 (b) 0 lt P(E) lt 1 (c) 0 le P(E) le 1 (d) 0 le P(E) lt 1