A function f(x) is, continuous in the closed interval [0,1] and differentiable in the open interval [0,1] prove that `f'(x_(1))=f(1) -f(0), 0 lt x_(1) lt 1`

Let f(x), be a function which is continuous in closed interval [0,1] and differentiable in open interval 10,1[ Show that EE a point c in]0,1[ such that f'(c)=f(1)-f(0)

If the function f(x) is continuous in the interval [-2, 2]. find the values of a and b where {:(f(x)=(sinax)/(x)-2," ,for "-2 le x lt0),(=2x+1," ,for "0 le x le1),(=2bsqrt(x^(2)+3)-1," ,for " 1 lt x le 2):}

Leg f be continuous on the interval [0,1] to R such that f(0)=f(1). Prove that there exists a point c in [(0,1)/(2)] such that f(c)=f(c+(1)/(2))

Let f:[0,1]rarrR (the set of all real numbers) be a function. Suppose the function f is twice differentiable, f(0)=f(1)=0 and satisfies f\'\'(x)-2f\'(x)+f(x) ge e^x, x in [0,1] If the function e^(-x)f(x) assumes its minimum in the interval [0,1] at x=1/4 , which of the following is true? (A) f\'(x) lt f(x), 1/4 lt x lt 3/4 (B) f\'(x) gt f(x), 0 ltxlt1/4 (C) f\'(x) lt f(x), 0 lt x lt 1/4 (D) f\'(x) lt f(x), 3/4 lt x lt 1

Let the function f(x)=3x^(2)-4x+5log(1+|x|) be defined on the interval [0,1]. The even extension of f(x) to the interval [0,1]. The even extension of f(x) to the interval [-1,1] is

If f be a continuous function on [0,1] differentiable in (0,1) such that f(1)=0, then there exists some c in(0,1) such that

If f be a continuous function on [0,1], differentiable in (0,1) such that f(1)=0, then there exists some c in(0,1) such that