Let `f(x)={{:(xe^(ax)",", x le0),(x+ax^2-x^3",",x gt 0):}` where a is postive constant . Find the interval in which f'(X) is increasing.

Find the interval in which f(x)=2x^(3)+3x^(2)-12x+1 is increasing.

Let f (x) { {:(1+x"," , 0 le x le 2),( 3-x"," ,2 lt x le 3):}: Find fof.

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.

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

Discuss the continuity of f(x) in [0, 2], where f(x) = {{:([cos pi x]",",x le 1),(|2x - 3|[x - 2]",",x gt 1):} where [.] denotes the greatest integral function.

Let f(x)=sin^(-1)((2phi(x))/(1+phi^(2)(x))) . Find the interval in which f(x) is increasing or decreasing.

Find the intervals in which f(x) = 2x^3+9x^2+12x+20 is increasing or decreasing.

if f(x) = {{:(x^2, x le 1),(ax + b ,x gt 1):} is differentiable at x = 1 . Find the values of a and b.

Find the intervals in which the function f given by f(x) = 2x^2 - 3x is increasing

If a function f(x) is defined as f(x) = {{:(-x",",x lt 0),(x^(2)",",0 le x le 1),(x^(2)-x + 1",",x gt 1):} then a. f(x) is differentiable at x = 0 and x = 1 b. f(x) is differentiable at x = 0 but not at x = 1 c. f(x) is not differentiable at x = 1 but not at x = 0 d. f(x) is not differentiable at x = 0 and x = 1