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if f(x) ={{:(-x=(pi)/(2),xle -(pi)/(2))...

`if f(x) ={{:(-x=(pi)/(2),xle -(pi)/(2)),(- cos x, -(pi)/(2)lt x ,le 0),( x-1, 0 lt x le 1),("in"x, x gt1):}`

A

f(x) is continuous at `x = - (pi)/(2)`

B

f(x) is not differentiable at x = 0

C

f(x) is differentiable x = 1

D

f(x) is differentiable at `x = -(3)/(2)`

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The correct Answer is:
D
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{:(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) = {{:(-2 sin x,"for",-pi le x le - (pi)/(2)),(a sin x + b,"for",-(pi)/(2) lt x lt (pi)/(2)),(cos x,"for",(pi)/(2) le x le pi):} . If f is continuous on [-pi, pi) , then find the values of a and b .

The following functions are continuous on (0, pi) (a) tan x (b) underset(0)overset(x)int t "sin"(1)/(t)dt (c) {{:(,-1,0 lt xle (3pi)/(4)),(,2sin((2)/(9)x),(3pi)/(4)lt xlt pi):} (d) {{:(,x sin x,0 lt x le (pi)/(2)),(,(pi)/(2)sin(pi+x),(pi)/(2)lt x lt pi):}

For each ositive integer n, define a function f _(n) on [0,1] as follows: f _(n((x)={{:(0, if , x =0),(sin ""(pi)/(2n), if , 0 lt x le 1/n),( sin ""(2pi)/(2n) , if , 1/n lt x le 2/n), (sin ""(3pi)/(2pi), if, 2/n lt x le 3/n), (sin "'(npi)/(2pi) , if, (n-1)/(n) lt x le 1):} Then the value of lim _(x to oo)int _(0)^(1) f_(n) (x) dx is:

f(x) = {{:((sqrt(1+kx)-sqrt(1-kx))/(x),if -1 le x lt 0),((2x+1)/(x-1),if 0 le x le 1):} at x = 0 .

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