`f(x)={(ae^x+be^(-x),-1lexle1),(cx^2,1lexle3),(2ax+c,3lexle4):}` ,`f'(0)+f'(2)=`

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

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

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

The function f(x) Is defined as follows : f(x)={{:(x^(2)+ax+b" , "0lexlt2),(3x+2" , "2lexle4),(2ax+5b" , "4ltxle8):} . If f(x) is continuous on [0,8], find the values of 'a' and 'b'.

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

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

If f(x)=[(x,xlt1),(x^(2),1lexle4),(8sqrt(x),xgt4):} the find f^(-1)(x) .

For what choices of a, b, c, if any, does the function f(x)={{:(ax^(2)+bx+x" , "0lexle1),(bx-c" , "1lexle2),(x" , "xgt2):} become differentiable at x = 1 and x = 2?