Find the average value of `mu` of the function f(x) over the indicated intervals:
(a) `f(x) = 2x^(2) + 1` over [0, 1]
(b) `f(x) = (1)/(x)` over [1, 2]
(c ) `f(x) = 3^(x)-2x + 3` over [0, 2]

Find the average value mu of the function f(x) = root(3)(x) over the interval [0, 1]

Find the average values of this functions: (a) f(x) = sin^(2)x " over " [0, 2pi] (b) f(x) = (1)/(e^(x) + 1) over [0, 2]

Find the greatest and the least values of the following functions on the indicated intervals : f(x) = sqrt((1-x)^(2)(1+2x^(2)))" on"[-1,1]

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].

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].

The value of b for which the function f(x)=x+cos x+b is strictly decreasing over R is: a) b =1

The value of b for which the function f(x)=x+cos x+b is strictly decreasing over R is: a) b =1

Verify lagranges mean value theorem for the following functions on the indicated intervals. Also,find a point c in the indicated interval: f(x)=x(x-2) on [1,3]f(x)=x(x-1)(x-2) on [0,(1)/(2)]

Verify the Lagrange's theorem for the following functions: (a) f(x) = x^(3) in [-1,1] (b) f(x) = sin x in [0,pi/2] ( c) f(x) = x^(n) in [-1,1], n in Z^(+) ( d) f(x) =2x^(2) - 7x + 10, x in [2,5]

Wht is the interval over which the function f(x)=6x-x^(2),x gt 0 is increasing ?