Find number of roots of f(x) where `f(x) = 1/(x+1)^3 -3x + sin x`

Find lim_(x to 1) f(x) , where f(x) = {{:(x + 1, x != 1),(0, x = 1):}}

If f (x)= [{:( cos x ^(2),, x lt 0),( sin x ^(3) -|x ^(3)-1|,, x ge 0):} then find the number of points where f (x) =f )|x|) is non-difierentiable.

if f(x) is differentiable function such that f(1) = sin 1, f (2)= sin 4 and f(3) = sin 9, then the minimum number of distinct roots of f'(x) = 2x cosx^(2) in (1,3) is "_______"

In f (x)= [{:(cos x ^(2),, x lt 0), ( sin x ^(3) -|x ^(3)-1|,, x ge 0):} then find the number of points where g (x) =f (|x|) is non-differentiable.

Find k if fog=gof where f(x)=3x-1 , g(x)=4x+k .

If (1+alpha)/(1-alpha),(1+beta)/(1-beta), (1+gamma)/(1-gamma) are the cubic equation f(x) = 0 where alpha,beta,gamma are the roots of the cubic equation 3x^3 - 2x + 5 =0 , then the number of negative real roots of the equation f(x) = 0 is :

If (1+alpha)/(1-alpha),(1+beta)/(1-beta), (1+gamma)/(1-gamma) are the cubic equation f(x) = 0 where alpha,beta,gamma are the roots of the cubic equation 3x^3 - 2x + 5 =0 , then the number of negative real roots of the equation f(x) = 0 is :

The number of points at which f(x)=[x]+|1-x|, -1lt x lt3 is not differentiable is (where [.] denotes G.I.F)

If f: R^+ to R , f(x) + 3x f(1/x)= 2(x+1),t h e n find f(x)

Let lim_(x to 0) f(x) be a finite number, where f(x) = ("sin" x + ae^(x) + be^(-x) + cln (1 + x))/(x^(3)) a,b,e in R