Let `g(t)=abs(t-1)-abs(t)+abs(t+1),forall " " t in R.`
Find `f(x)=max{g(t):-3/2 le t le x},forall x in ((-3)/2,infty)`.]

f(x)=abs(x-1)+abs(x-2), -1 le x le 3 . Find the range of f(x).

{:(f(x) = cos x and H_(1)(x) = min{f(t), 0 le t lt x},),(0 le x le (pi)/(2) = (pi)/(2)-x,(pi)/(2) lt x le pi),(f(x) = cos x and H_(2) (x) = max {f(t), o le t le x},),(0 le x le (pi)/(2) = (pi)/(2) - x","(pi)/(2) lt x le pi),(g(x) = sin x and H_(3)(x) = min{g(t),0 le t le x},),(0 le x le (pi)/(2)=(pi)/(2) - x, (pi)/(2) le x le pi),(g(x) = sin x and H_(4)(x) = max{g(t),0 le t le x},),(0 le x le (pi)/(2) = (pi)/(2) - x, (pi)/(2) lt x le pi):} Which of the following is true for H_(1) (x) ?

Let f(x) = x^(3) - x^(2) + x + 1 and g(x) = {{:(max f(t)",", 0 le t le x,"for",0 le x le 1),(3-x",",1 lt x le 2,,):} Then, g(x) in [0, 2] is a. continuous for x in [0, 2] - {1} b. continuous for x in [0, 2] c. differentiable for all x in [0, 2] d. differentiable for all x in [0, 2] - {1}

Let f(x) = 1 + 4x - x^(2), AA x in R g(x) = {max {f(t), x le t le (x + 1) 0 le x lt 3 min {(x + 3) 3 le x le 5} Verify continuity of g(x), for all x in [0, 5]

Let f and g be real - valued functions such that f(x+y)+f(x-y)=2f(x)*g(y), forall " x , y " in R. Prove that , if f(x) is not identically zero and abs(f(x)) le 1, forall x in R, then abs(g(y)) le 1, forall y in R.

Let f(x) = sin x and " g(x)" = {{:(max {f(t)","0 le x le pi},"for", 0 le x le pi),((1-cos x)/(2)",","for",x gt pi):} Then, g(x) is

If g(t) = 1- 4t^2 , find g’(1).

If f(x) is continuous such that abs(f(x)) le 1, forall x in R " and " g(x)=(e^(f(x))-e^(-abs(f(x))))/(e^(f(x))+e^(-abs(f(x)))), then range of g(x) is

f(x) = cos x and H(x) = {{:(min{f(t), 0 le t lt x},0 le x le (pi)/(2)) ,( (pi)/(2)-x,(pi)/(2) lt x le 3):} then

If x = g(t) and y = f(t) , find (d^2y)/(dx^2) as a function of t.