Using Largrange's mean value theorem, prove that,
`b^(n)-a^(n) gt n a^(n-1)(b-a)" when " b gt a `.

Using Largrange's mean value theorem, prove that, tan^(-1)x lt x

Using Lagrange's mean value theorem prove that, e^(x) gt 1 +x

Using Lagranges mean value theorem, prove that |cosa-cosb|<=|a-b|dot

Using Lagranges mean value theorem, prove that |cosa-cosb|<=|a-b|dot

Using Lagrange's mean value theorem prove that, log (1+x) lt x ("when " x gt 0)

If a > b >0, with the aid of Lagranges mean value theorem, prove that n b^(n-1)(a-b) < a^n -b^n < n a^(n-1)(a-b) , if n >1. n b^(n-1)(a-b) > a^n-b^n > n a^(n-1)(a-b) , if 0 < n < 1.

Using Lagranges mean value theorem, prove that (b-a)/bltlog(b/a)lt(b-a)/a, where 0ltaltbdot

Using Largrange's mean value theorem f(b)-f(a)=(b-a)f'(c ) find the value of c, given f(x)=x(x-1)(x-2), a=0 and b=(1)/(2) .

Prove that 1^(2) +2^(2)+ ….+n^(2) gt (n^(3))/(3) n in N

Using binomial theorem, prove that 2^(3n)-7n-1 is divisible by 49 , where n in Ndot