Let ` f(x) ={{:( ( 2x-1) "," ,"when" , x ge 1),( x",", "when", 0 le x le 1):}`
Show that f(x) is continuous but not differentiable at x =1

If f (x) ={{:(x",", "when " 0 lt x lt 1),(2-x",", "when" 1 lt x le 2):} then f'(1) is equal to-

If f(x)={:{(x^2", "x le 1),( x^2-x+1"," x gt1):} then show that f(x) is not differentiable at x=1 .

If f(x)={:{(x^2", "x le 1),( x^2-x+1"," x gt1):} then show that f(x) is not differentiable at x=1 .

Let f(x)={{:(2x+1,"when"1 le x le 2),(x^(2)+1,"when"2 le x le 3):} . Show that int_(1)^(3)f(x)dx=(34)/(3) .

Let f (x) = {{:((1+sin x )",", when, 0 le x lt (pi)/(2)), (" " 1",", when, x lt 0 ):} Show that f ' (0) does not exist.

A function is defined as f(x) = {{:(e^(x)",",x le 0),(|x-1|",",x gt 0):} , then f(x) is(a) continuous at x = 0 ,(b) continuous at x = 1 (c)differentiable at x = 0

Let f(x) = [{:( sin (pi)/(2) x "," , 0 le x le 1) , (3 - 2x "," , x ge 1):} then

Let f(x) = {:{(x+a, x le1),(ax^2 + 1, x > 1):} . Show that f is continuous at 1. Find 'a' so that it is derivable at 1.

Let f(x) = {:(1/(x^2) , : |x| ge 1),(alphax^2 + beta , : |x| < 1):} . If f(x) is continuous and differentiable at any point, then

Show that the funcation f(x)={{:(3x-2",", " when ", x le0),(x+1",", " when ", x gt 0):} is discontinuous at x =0