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]`Which of the following is true for `0 lt x lt 1` ? (A) `0 lt f(x) lt oo` (B) `-1/2 lt f(x) lt 1/2` (C) `-1/4 lt f(x) lt 1` (D) `-oo lt f(x) lt 0`

Let f[0, 1] -> R (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) leq e^x, x in [0, 1] .Which of the following is true for 0 lt x lt 1 ?

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

Find the inverse of each of the following functions : f(x) = {{:(x"," -oo lt x lt 1),(x^(2)"," 1 le x le 4),(2x"," 4 lt x lt oo):}

Consider the function f:(-oo, oo) -> (-oo ,oo) defined by f(x) =(x^2 - ax + 1)/(x^2+ax+1) ;0 lt a lt 2 . Which of the following is true?

Prove that the greatest integer function defined by f(x) =[x] ,0 lt x lt 2 is not differentiable at x=1

Given that f is a real valued differentiable function such that f(x) f'(x) lt 0 for all real x, it follows that

Find k, if the following is p.d.f or r.v.f X. f(x) = {{:(kx^(2)(1-x), 0 lt x lt 1),(0, "otherwise"):}

There exists a function f(x) satisfying 'f (0)=1,f(0)=-1,f(x) lt 0,AAx and

If f(x)={:{(x", for " 0 le x lt 1),(2", for " x=1),(x+1", for " 1 lt x le 2):} , then f is