Does there exist a point `c(in)(0,pi/2)` such that f' (c)= 0 where f(x) = sin 2x`?

Can we find a point c^(in) (-pi/2,pi/2) such that f'© = 0 where f(x) = |sinx|

If f(x) is continuous function in [0,2pi] and f(0)=f(2 pi ), then prove that there exists a point c in (0,pi) such that f(x)=f(x+pi).

Find a point c^(in) (-1,2) such that f'© = (f(2) - f(-1))/(2-(-1) where f(x) = x^2

Let f(x) = x(x-1)(x-2),x in [0,2]. Prove that 'f' satisfies the conditions of Rolle's Theorem and there is more than one 'c' in (0,2) such that f'(C ) = 0

Let f:[0,1]toR (the set of all real numbers ) be a function. Suppose the function f is twice differentiable,f(0)=f(1)=0 and satisfies f''(x)-2f'(x)+f(x)gee^(x),x in [0,1] Consider the statements. I. There exists some x in R such that, f(x)+2x=2(1+x^(2)) (II) There exists some x in R such that, 2f(x)+1=2x(1+x)

Given f'(1)=1and f(2x)=f(x)AAxgt0.If f'(x) is differentiable, then there exists a number c in (2,4) such that f''(c ) equal

In [0,pi/2] the function f(x) = log sin x is

Suppose that f(x) is continuous in [0, 1] and f(0) = 0, f(1) = 0 . Prove f(c) = 1 - 2c^(2) for some c in (0, 1)

Area bounded by y=f^(-1)(x) and tangent and normal drawn to it at points with abscissae pi and 2pi , where f(x)=sin x-x is

Verify Rolle's theorem for the following functions in the given intervals and find a point (or point) where the derivative vanishes : f(x) = sin^4x + cos^4 x in [0,pi/2]