A continuous function f(x) is defined as follows :
`f(x)={{:(x,"when "xlt1),(ax+bx^(2),"when "1lexle2),(2x^(2),"when "xgt2):}`
Find the values of a and b.

A function f (x) is defined as follows: f(x)={{:(x^(4)-3,"when " xle2),(x^(3)+5,"when "xgt2):} Prove that the function f (x) is continuous at x = 2 .

A function f(x) is defined as follows : " "f(x)={:{(x-1",","when "x gt 0),(-(1)/(2)",","when "x = 0),( x+1",","when "xlt0):} Draw the graph of the function f(x). From the graph find the value of f(x) at x=-(1)/(2) and examine whether f(x) is continuous at x = 0.

A function f (x) is defined as follows : f(x)={{:(3x+1,"for "xle1 ),(3-ax^(2),"for " xgt1):} If f (x) is continuous at x = 1 , find the value of a.

Let f(x)={{:(2x+1,"when "xlt2),(k+2,"when "x=2),(4x-3,"when "xgt2):} Find the value of k for which f (x) is continuous at x = 2 .

A function f (x) is defined as follows : f(x)={{:(2-x",","when "xlt1),(x^(2)-3x",","when "xge1):} Examine the differentiability of the function f (x) at x= 1 , hence state whether f (x) is continuous at x = 1 or not .

A function f (x) is defined as follows: f(x)={{:(x+2,"when "xlt2),(x^(2)-1,"when "xge2):} Show that f (x) is discontinuous at x = 2 and the jump of the function at this point is -1 .

A function f (x) is defined as followes: f(x)={{:(xsin.(1)/(x),"when " xne0),(0,"when " x=0):} Show that f (x) is continuous at x =0 .

A function f (x) is defined as follwos : f(x){:(=2x+1," when "xle 1),(= 3-x," when "xgt1):} Examine whether underset(xrarr1)"lim"f(x) exists or not.

A function f(x) is defined as follows : f(x)=-x," when "x lt0 =2," when "0 lex lt2 =4-x," when "x ge2 Draw the graph of the function and from the graph examine whether f(x) is continuous at x = 0 and x = 2 or not.

A function is defined as follows : f(x) = {((x^(2))/(2),"when " 0 le x lt 1),(2x^(2) - 3x + (3)/(2),"when " 1 le x le 2):} Discuss the existence of f'(1).