If `sqrt( 2 + x - x^2 ) > x - 4` then find the interval of x

Find the value of c, of mean value theorem.when (a) f(x) = sqrt(x^(2)-4) , in the interval [2,4] (b) f(x) = 2x^(2) + 3x+ 4 in the interval [1,2] ( c) f(x) = x(x-1) in the interval [1,2].

Verify Rolle's theorem for the function f(x) = 2x^(3) + x^(2) - 4x - 2 in the interval [-(1)/(2) , sqrt2] .

If x=(4sqrt(2))/(sqrt(2)+1) , then find the value of (x+2)/(x-2)-(x+2sqrt(2))/(x-2sqrt(2)) .

Given sqrt(4-x^(2))>=(1)/(x), find the interval in which x lies.

If log_((1)/(sqrt(2)))(x-1)>2, then x lies is the interval

Using Lagrange's theorem , find the value of c for the following functions : (i) x^(3) - 3x^(2) + 2x in the interval [0,1/2]. (ii) f(x) = 2x^(2) - 10x + 1 in the interval [2,7]. (iii) f(x) = (x-4) (x-6) in the interval [4,10]. (iv) f(x) = sqrt(x-1) in the interval [1,3]. (v) f(x) = 2x^(2) + 3x + 4 in the interval [1,2].

If x=2+sqrt(3) , then find the value of x^(4)-4x^(3)+x^(2)+x+1 .