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)`.]

Let f(t)=|t-1|-|t|+|t+1|, AA t in R . Find g(x) = max {f(t):x+1letlex+2}, AA x in R . Find g(x) and the area bounded by the curve y=g(x) , the X-axis and the lines x=-3//2 and x=5 .

Let f(x) = (1-x)^(2) sin^(2)x+ x^(2) for all x in IR and let g(x) = int_(1)^(x)((2(t-1))/(t+1)-lnt) f(t) dt for all x in (1,oo) . Which of the following is true ?

{:(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_(3) (x) ?

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 3min {(x + 3), 3 le x le 5} Verify conntinuity of g(x), for all x in [0, 5]

If f(x)=t^(2)+(3)/(2)t , then f(q-1)=

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

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

Let f(x) = int_(0)^(x)(t-1)(t-2)^(2) dt , then find a point of minimum.

Let f (x) =(1)/(x ^(2)) int _(4)^(x) (4t ^(2) -2 f '(t) dt, find 9f'(4)