If `f(x)={{:(x",",0lexle1),(2-e^(x-1)",",1ltxle2),(x-e",",2ltxle3):}` and `g'(x)=f(x), x in [1,3]`, then```

Let f(x)={{:(3^x","-1lexle1),(4-x","1ltxle4):} then

If f(x)={{:(3x^(2)+12x-1",",-1lexle2),(37-x",",2ltxle3):} , Then

Let function f(x) is defined in [-2,2] as f(x)={{:({x}",",-2lexle-1),(|sgnx|",",-1lexle1),({-x}",",1ltxle2):} where {x} and sgn x denotes fractional part and signum functions, respectively. Then the area bounded by the graph of f(x) and x-axis is

Let f(x)={(x+1, -1lexle0),(-x,0ltxle1):} then

Consider two function y=f(x) and y=g(x) defined as f(x)={{:(ax^(2)+b,,0lexle1),(bx+2b,,1ltxle3),((a-1)x+2c-3,,3ltxle4):} and" "g(x)={{:(cx+d,,0lexle2),(ax+3-c,,2ltxlt3),(x^(2)+b+1,,3gexle4):} lim_(xrarr2) (f(x))/(|g(x)|+1) exists and f is differentiable at x = 1. The value of limit will be

Consider two function y=f(x) and y=g(x) defined as f(x)={{:(ax^(2)+b,,0lexle1),(bx+2b,,1ltxle3),((a-1)x+2c-3,,3ltxle4):} and" "g(x)={{:(cx+d,,0lexle2),(ax+3-c,,2ltxlt3),(x^(2)+b+1,,3gexle4):} Let f be differentiable at x = 1 and g(x) be continuous at x = 3. If the roots of the quadratic equation x^(2)+(a+b+c)alphax+49(k+kalpha)=0 are real distinct for all values of alpha then possible values of k will be

Let f(x)={{:(5x-4",",0ltxle1),(4x^(3)-3x",",1ltxlt2.):} Find lim_(xrarr1)f(x).

If f(x) = {{:(e ^(x),,"," 0 le x lt 1 ,, ""), (2- e^(x - 1),,"," 1 lt x le 2,, and g(x) = int_(0)^(x) f(t ) dt","),( x- e,,"," 2lt x le 3 ,, ""):} x in [ 1, 3 ] , then

If f(x)={{:(1-x",",0lexle2),(x-1",",2lexle4),(1",",4lexle6):} then find, f(0)+f((1)/(2))+f(1)+f((45)/(18)):