Prove that `f(x)=x-sinx` is an increasing function.

Prove that the function f(x)=x-sinx is increasing on the real line . Also discuss for the existence of local extrema.

Let f be a function defined on [a, b] such that f ′(x) gt 0, for all x in (a, b). Then prove that f is an increasing function on (a, b).

Let f: R->R be a differentiable function for all values of x and has the property that f(x)a n df^(prime)(x) has opposite signs for all value of xdot Then, (a) f(x) is an increasing function (b) f(x) is an decreasing function (c) f^2(x) is an decreasing function (d) |f(x)| is an increasing function

Let f:[1,oo)->R and f(x)=x(int_1^xe^t/tdt)-e^x .Then (a) f(x) is an increasing function (b) lim_(x->oo)f(x)->oo (c) fprime(x) has a maxima at x=e (d) f(x) is a decreasing function

Let f be the function f(x)=cosx-(1-(x^2)/2)dot Then f(x) is an increasing function in (0,oo) f(x) is a decreasing function in (-oo,oo) f(x) is an increasing function in (-oo,oo) f(x) is a decreasing function in (-oo,0)

If f(x)=x^3+b x^2+c x+d and 0 le b^2 le c , then a) f(x) is a strictly increasing function b)f(x) has local maxima c) f(x) is a strictly decreasing function d) f(x) is bounded

Prove that f(x)=(sinx)/x is monotonically decreasing in [0,pi/2]dot Hence, prove that (2x)/pi

Prove that the function are increasing for the given intervals: f(x)=e^x+sinx ,x in R^+

Prove that the function are increasing for the given intervals: f(x)=sinx+tanx-2x ,x in (0,pi/2)

Prove that the function are increasing for the given intervals: f(x)=secx-cos e cx ,x in (0,pi/2)